Die Methode Der Nichtlokalen Effektiven Wirkung in Höherdimensionalen Raumzeitmodellen / the Method of the Nonlocal Effective A

Die Methode der nichtlokalen eﬀektiven 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 2003 Dekan: Prof. Dr. Rolf Schneider Leiter der Arbeit: Prof. Dr. Hartmann Römer Referent: Prof. Dr. Hartmann Römer Korreferent: Prof. Dr. Jochum van der Bij Tag der mundlic¨ hen Prufung:¨ 23.10.2003 The Method of the Nonlocal Effective Action in Higher-Dimensional Spacetime Models

Andreas Rathke

Contents

1 Effective methods in braneworld dynamics 7 1.1 Introduction ...... 7 1.1.1 The role of the nonlocal effective action ...... 10 1.1.2 Overview ...... 11 1.2 The Randall-Sundrum models ...... 12 1.3 Effective actions for warped braneworlds ...... 14 1.3.1 Kaluza-Klein description ...... 15 1.3.2 Holographic description ...... 18 1.3.2.1 One-brane effective 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 effective action of brane-localized fields and the methods of its calculation ...... 26 2.2.1 The structure of the braneworld effective action ...... 26 2.2.2 The role of radion fields ...... 29 2.3 Two-brane Randall-Sundrum model: the final answer for the two-field braneworld action ...... 30 2.4 The effective equations of motion ...... 34 2.5 The recovery of the braneworld effective 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-field 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 effective action ...... 53 2.8.1 Small interbrane distance ...... 55 2.8.2 Large interbrane distance 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 Eﬀective action of brane-localized ﬁelds vs. Kaluza-Klein reduction . 63 3.2 The recovery of the Kaluza-Klein tower ...... 64

5 6

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 eﬀective action . . 67 3.2.4 The graviton eﬀective action in the diagonalization approximation ...... 71 3.3 Phenomenological digression: radion-induced graviton oscillations . . 73 3.3.1 Overview ...... 73 3.3.2 Gravitational waves on the Σ+-brane ...... 75 3.3.3 Quantum oscillations — an analogy ...... 76 3.3.4 Gravitational-wave oscillations on the Σ+-brane ...... 78 3.3.5 High-amplitude RIGO’s on the Σ+-brane from M-theory . . . 79 3.3.6 Graviton oscillations on the Σ -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-de Sitter space and the geometrical setting of the RS models 99

B Diagonalization of the kinetic and mass terms 101

Bibliography 105 1

Eﬀective methods in braneworld dynamics

1.1 Introduction

Among the most promising candidates for a unified 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 fields associated with the ends of a string reside on the branes. Gauge fields, rather than spreading through the entire ten-dimensional spacetime of superstring theory, are thus confined 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 fields 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 compactified flat higher dimensions, 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 massive graviton states, only observable at higher energies. It is imaginable that the compactification is large enough in order to produce effects at accelerators of the next generation [4]. In a different scenario [7, 8]. The higher-dimensional manifold does no longer have to “factorize”1 into our four-dimensional world and the additional dimensions. In particular, one can assume that the bulk space has an anti-de Sitter (AdS) structure transverse to the branes. The brane showing the behavior of effective four- 2 dimensional gravity (“our Universe”) is put at an Z2-orbifold point in the bulk

1The term non-factorizable geometry is used in the context of higher-dimensional spacetime models diﬀerently from its actual geometrical meaning. A higher-dimensional 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 diﬀerent 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

7 8 1 Effective methods in braneworld dynamics space. Consequently, the bulk metric has a reflection-symmetry with respect to the position of the brane. The metric of AdS space yields an effective 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] (commonly 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 effective 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 de Sitter 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 effective 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 effects [19], though it is still under dispute if these models exhibit a realistic behavior in the limit of strong gravitational fields [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, neglecting 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 effect of bulk gravity on the brane can be replaced by a four-dimensional conformal field theory with an ultraviolet cutoff via the anti-de Sitter/conformal-field-theory (AdS/CFT) correspondence (cf. [24]). This provides a direct technical connection between brane-world scenarios and string theory. Only one-brane models have a zero-mode approximation that corresponds to four-dimensional gravity without additional fields (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 offers a potential solution to the hierarchy problem – the huge ratio between the magnitudes of the electroweak and gravitational couplings. In models with compactified dimensions, this is due to the fact that gravity propagates through the bulk and consequently is weakened by a factor of the inverse

1 M. In the following we only consider the special case of a S /Z2 orbifold, i. e. reflection symmetry applied to a circle. We call this case a 2 orbifold and the points of the circle with respect to which Z the reflection 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 effective 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 compactifications on a circle. They might therefore offer 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 modifications 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 inflationary scenarios driven by scalar fields on the brane [34, 35] or the trace anomaly of the matter fields on the brane [36, 37]. The latter analysis has heavily drawn on the technical benefits of the AdS/CFT correspondence. On the other hand, the influence of bulk fields on the cosmological evolution on the branes has been discussed. These bulk fields tend to drive the branes closer to each other, finally 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 cosmic microwave background radiation. These models have, however, also been heavily criticized [42, 43].

In models with compactification on a circle one has not yet succeeded to generate viable inflationary scenarios in which the inflation on the brane is driven by the modulus field [44], i. e. the field describing the size of the extra dimension. However, a recent study has shown that it might be possible to generate inflation on the brane without any additional scalar field in the bulk or on the brane in a modified RS two-brane model [45]. This is achieved using the scalar modulus of the extra dimension, the “radion”, as an inflaton. The potential that provides the necessary energy for the expansion is part of the tension of the brane on which inflation occurs. During inflation on the brane, the branes move away from each other. The model of [45] provides an inflaton potential which tends to a fixed value for infinite 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 inflation [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 inflation.

There are also attempts to explain the creation of braneworlds by the methods of quantum cosmology. In this approach the creation 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 eﬀective action

In the context of this rapidly evolving field of research we want to propose a new method of deriving the four-dimensional effective 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 spacetime 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 description also provides the framework for a convenient comparison between the cosmological observations in our Universe and the dynamics of a braneworld model. There exist various prescriptions for obtaining the effective degrees 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 infinite. 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 effective 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 effective four-dimensional equations of motion. Another popular approach for the analysis of braneworld cosmologies is the use of moduli effective actions [53, 54]. In this method the geometry of the branes is chosen as fixed 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 five-dimensional degrees of freedom taken into account are the moduli-fields of the higher dimensions. In the particular case of an Anti-de Sitter bulk another possibility for obtaining the on-brane effective dynamics opens up by making use of the AdS/CFT correspondence [55]. It relates the gravitational dynamics on an Anti-de Sitter space to the dynamics of a conformal field 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 field theory. Unfortunately, this in principle clear procedure remains technically involved. Not all parameters necessary to describe the first-order CFT contribution to the on-brane dynamics could yet be obtained in this approach. Also the effect of a second brane cutting off 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 effective 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 1.1 Introduction 11 both, the RS two-brane and the RS one-brane model as an expansion up to second order in powers of the curvature, in which the coefficients of the curvature terms will be in general nonlocal functions of the d’Alembert operator. It is found that the nonlocality of the action does not obstruct using the action in a cosmological context. Relocalization and approximation schemes exist which allow cosmological insights beyond the ones gained with other methods.

1.1.2 Overview The layout of our considerations is the following: this introductory part will continue with a description of the background solutions of the Randall Sundrum two-brane and one-brane models. The rest of the first chapter gives an introduction to the two different methods of computing the effective action for the RS model: the Kaluza- Klein description and the holographic description. These two methods deserve some attention in our context because their results considerably facilitate the understanding of the findings obtained later by means of our calculations of the nonlocal effective action. The second part of this work then presents and derives our method in detail. After a theoretical overview on the method in Sec. 2.2.1, we present the main result of our work: the two-field action for the RS two-brane model in Sec. 2.3. The Secs. 2.4 to 2.6 are devoted to the derivation of this result while the following Secs. 2.7 and 2.8 discuss the particle content and some interesting limiting cases and simplifications of the general action. A detailed survey of the contents of this part is given in Sec. 2.1. The organization of Chap. 2 follows in large part [56], for an abridged discussion see [57]. The third part will be concerned with the interpretation of the nonlocal effective action. The understanding of our results is considerably facilitated by interpreting them in terms of the two methods described in the introduction. We start in Sec. 3.1 by a comparison of the nonlocal-action approach and the Kaluza-Klein description of the RS two-brane model. These, rather theoretical, considerations are significantly enriched in Sec. 3.2 by the explicit recovery of the KK tower of gravitons from the nonlocal effective action. The demonstration of the correspondence between the KK description and the nonlocal effective action will be followed by a chapter lying a little outside of the main scope of: Sec. 3.3 analyzes the phenomenologically relevant effect of radion-induced gravitational wave oscillations (RIGO’s) which might even become the first experimental signature of brane worlds. This effect could in principle already have been discovered from the KK description of braneworld gravity and therefore does not take full advantage of the nonlocal action method. We nevertheless decided to include it because it is an application of the particle interpretation of the nonlocal action and also due to its phenomenological relevance. The material of Secs. 3.2 and 3.3 is based on [58]. Various aspects of these topics have already been discussed in [59] and [60]. In Sec. 3.4 the main line of Chap. 3 is resumed when the equivalence of the nonlocal effective action and the holographic action for the RS models will be demonstrated. By the comparison for the one-brane model we fix an open parameter in the holographic action which previously limited the utility of the holographic method. In order to describe the relation in the two-brane model we extend the holographic method to the two-brane setup in the limit of small brane separation. 12 1 Effective methods in braneworld dynamics

The final part, Chap. 4, sketches some cosmological implications of the nonlocal effective action, underlining its versatile applicability. In particular, it discusses our results with regard to a possible mechanism of braneworld inflation with repelling (“thick”) branes. We will also give an outlook on applications and possible generalizations of the new method. In particular we will suggest the nonlocality of braneworld actions as a possible effect to explain the present-day cosmic acceleration and the galactic rotation curves. We also outline a way in which the nonlocal braneworld action can be used as a tool to investigate aspects of the AdS/CFT correspondence.

1.2 The Randall-Sundrum models

A brane can be approximated as a δ-distribution-like source of the gravitational field if one considers distances much larger than the brane-thickness. Then a brane can be treated as a hypersurface in the bulk space. In the analysis of the present work we will content ourselves with branes of co-dimension one. In the simplest case the brane is completely characterized by only one parameter, the brane tension σ, which describes the energy density per unit three-volume. In general, we will also add an action for four-dimensional matter residing on the brane. In the RS setups [7, 8], the braneworld effective action is induced by the five- dimensional bulk spacetime. The starting point is the action of the five-dimensional gravitational field with the metric GAB(x, y), A = (µ, 5), µ = 0, 1, 2, 3, of the bulk spacetime xA = (xµ, y), x5 = y, and matter fields φ confined to the two branes Σ ,

S[ G, g, φ ] = S5[ G ] + S4[ G, g, φ ] . (1.1) The bulk-space part of the action is given by the usual Einstein-Hilbert action with 5 curvature scalar R and a cosmological constant Λ5

1 5 5 S5[ G ] = d x √G R(G) 2Λ5 . (1.2) 16πG5 − Z5 M

In (1.2) √G denotes the square root of the determinant of the metric and G5 is the ﬁve-dimensional gravitational constant. The brane-part of the action is

4 1/2 1 S4[ G, g, φ ] = d x Lm(φ, ∂φ, g) g σ + [K] . (1.3) Σ − 8πG5 X Z where the branes are enumerated by the index and carry induced metrics g (x), µν matter-ﬁeld Lagrangians Lm(φ, ∂φ, g) and four-dimensional cosmological constants σ .3 The Einstein-Hilbert bulk action (1.2) is accompanied by terms in (1.3) containing the jumps of the extrinsic curvature traces [K] associated with both sides of 4 each brane [62] . The bulk cosmological constant Λ5 is chosen to be negative and,

3Matter fields in the bulk have been discussed in the context of the RS model at various instances [61] but will remain outside the scope of our present discussion. 4 The extrinsic curvature Kµν is defined as a projection on the brane of the tensor AnB with the ∇ outward unit normal nB , i.e. the normal pointing from the bulk to the brane. With this definition the normals on the two sides of the brane are oppositely oriented and the extrinsic curvature jump [Kµν ] actually equals the sum of the so-defined curvatures on both sides of the brane. 1.2 The Randall-Sundrum models 13 therefore, capable of generating an AdS geometry, while the brane cosmological constants play the role of brane tensions and, depending on the model, can be of either sign. The field equations for this action are obtained by the standard variational procedure. In the bulk these are the five-dimensional Einstein equations. Variations with respect to the four-dimensional induced metrics on the branes give rise to the Israel junction conditions [63].5 In the RS two-brane setup, the fifth dimension labeled by the coordinate y, d y d, has the topology of a circle with an orbifold Z -identification of − ≤ ≤ 2 points y and y. The branes are located at antipodal fixed points of the orbifold, − y = y , y+ = 0, y = d. They are empty, i. e. Lm(φ, ∂φ, gµν ) = 0, and their | −| 6 tensions have opposite sign and are fine-tuned to the values of Λ5 and G5, 6 3 Λ5 2 , σ+ = σ = . (1.4) ≡ −l − − 4πG5l The model then admits a solution of the Einstein equations with an AdS metric in the bulk (l is its curvature radius),

2 2 2 y /l µ ν ds = dy + e− | | ηµνdx dx , (1.5)

0 = y+ y y = d , ≤ | | ≤ − with a flat induced metric η = diag[ 1, 1, 1, 1] on both branes [7]. The metric on µν − the negative tension brane is rescaled by the “warp factor” a2 exp( 2d/l), provid- ≡ − ing a possible solution for the hierarchy problem [7] by exponentially weakening the effective gravitational constant on the Σ -brane (cf. Sec. 2.6.1 below). With the fine − tuning (1.4) this solution exists for arbitrary brane separation d. If the fine-tuning (1.4) does not hold, the geometry of the branes will be de Sitter or anti-de Sitter (cf. e. g. [34, 65]). The RS one-brane model [8] is derived from this setup by removing the negative- tension brane. Maintaining the fine-tuning relation for the positive tension brane,

3 σ+ = , (1.6) 4πG5l one still has the metric (1.5) as a solution of the Einstein equations. In contrast to the two-brane setup the extra-dimension is no longer compact but inﬁnite. One can consider the RS one-brane model as the decompactiﬁcation limit of the two-brane

5Variations of the worldsheet of each brane give rise to the equations of motion for the radion associated with the brane. These variations will not be considered in the following as the radion equations of motion can also be obtained from the trace of the Israel junction conditions. cf. below Sec. 2.4. 6 The orbifold symmetry is motivated by the analogy to M-theory compactifications and by the unease of having an interval as the topology of a spacetime dimension. See, however, [64] for a list of string-theory realizations of spacelike dimensions which have interval topology. Choosing an interval as the topology of the fifth dimension would only result in changing the fine-tuning relation of (1.4) to σ = 3/(8πG5l). Therefore the introduction of the orbifold has no influence on the basic procdures we will describe. 2 4 The dimensions of the quantities appearing in (1.4) are: [Λ5] = mass , [σ] = mass and [G5] = mass−3. 14 1 Effective methods in braneworld dynamics model which is obtained by sending the negative-tension brane to infinity, y . − → ∞ Some aspects of the causal structure of the RS models are discussed in appendix A. In the following we will consider the description of the dynamics of small perturbations of the RS background solutions. The particle physics phenomenology of the RS models has been considered extensively in the literature, see [66, 67] for an introduction and a guide to the original papers.

1.3 Eﬀective actions for warped braneworlds

The efficient way describing the braneworld scenario is the method of the effective action. In general, this notion is very ambiguous because its precise meaning ranges from the generating functional of one-particle irreducible diagrams in local field theory to the low-energy effective action of target-spacetime fields in string theory. These definitions, apparently, have in common that the effective action manifestly describes the dynamics of the observables and, simultaneously, incorporates in implicit form the effect of invisible degrees of freedom, which are integrated (or traced) out. This type of description becomes actually indispensable in string-theoretical, Kaluza-Klein, and braneworld contexts when the observables turn out to be very different from the fundamental degrees of freedom whose dynamics underlie the effective “visible” dynamics. The situation is characteristic of the old Kaluza-Klein and new (braneworld) pictures because the “visible” fields φ(x) are four-dimensional in contrast to the fundamental fields Φ(x, y) in the multi-dimensional spacetime, which depend on the “visible” (four-dimensional) coordinates x and the coordinates of extra dimensions y. An- other example are the duality relations of the Anti-de Sitter/conformal-field-theory correspondence (AdS/CFT-correspondence) between the bulk- and boundary-theories widely celebrated by string physicists [55]. In these two cases, however, the four-dimensional fields φ(x) originate from Φ(x, y) by two very different procedures and their effective actions differ essentially. In the Kaluza-Klein formalism an infinite tower of KK modes φ(x) = φ (x) { n } arises as the coefficients of the expansion of Φ(x, y) in a certain complete set of harmonics on the y-space. Correspondingly, its effective action is just the original action of the fundamental field S[ Φ(x, y) ] rewritten in terms of φ (x) . This type { n } of action was constructed for the two-brane Randall-Sundrum scenario in [68, 69]. A good review of its particle phenomenology can be found in [66]. The KK description of extra-dimensional models and, in particular, of the RS two-brane model will be discussed in more detail in the next section. In contrast to this, in the holographic formalism of the AdS/CFT-correspondence 7 φ(x) roughly turns out to be the value of Φ(x, y) at the boundary, φ(x) = Φ(x, ybound). In the braneworld RS model, where the boundary is associated with the brane Σ located at ybrane, this identification has lead to the insight that the localization of the graviton zero-mode [8] and the recovery of the four-dimensional Einstein gravity on the brane [8, 21, 22] can be interpreted in terms of the AdS/CFT-correspondence

7Some of the subtleties of this identiﬁcation associated with the asymptotic properties of the AdS-spacetime boundary and the conformal structure of bulk and brane operators, will be addressed in Sec. 1.3.2.1. See also [70, 71]. 1.3 Effective actions for warped braneworlds 15

[24, 23, 36]. This conclusion was reached in the language of the effective action of the brane field — the four-dimensional metric induced on the brane from the bulk geometry. Thus, the value of the field at the respective brane, φ(x) = Φ(x, ybrane) becomes an appropriate variable to describe a braneworld scenario. Its effective action Seff [ φ(x) ] is obtained from the fundamental action S[ Φ ] by a less trivial procedure as in the case of a Kaluza-Klein reduction: one substitutes in S[ Φ ] a solution of the classical bulk equations of motion for Φ(x, y), Φ = Φ[ φ(x) ], which is parameterized by the boundary values of the bulk field at the branes, that is

Seff [ φ(x) ] = S Φ[φ(x) ] . This construction obviously generalizes to the case of I several branes ΣhI , enumeratedi by the index I, and the set of brane fields φ = Φ(ΣI). Such a definition corresponds to the tree-level approximation for the quantum effective action

exp iSeff [ φ ] = DΦ exp iS[ Φ ] , (1.7) Z Φ(Σ) = φ where the functional integration over the bulk fields runs subject to the brane boundary conditions. The scope of this formula is very large because it arises in very different contexts. In particular, its Euclidean version (iS S ) un- → − Euclid derlies the construction of the no-boundary wavefunction in quantum cosmology [72]. Semiclassically, in the braneworld scenario, it represents a Hamilton-Jacobi functional, and its evolutionary equations of motion in the “time” y can be interpreted as renormalization-group equations [73, 53]. It also forms the basis of the effective action formulation of the AdS/CFT-correspondence principle between supergravity theory on an AdS S5 background and the superconformal field theory 5 × (super-Yang-Mills) on its infinitely remote boundary [55, 23, 36, 37, 74, 75]. The holographic derivation of an effective four-dimensional action for the RS one-brane model will be described in Sec. 1.3.2.1.

1.3.1 Kaluza-Klein description

The most popular method to derive the eﬀective four-dimensional dynamics in higher-dimensional models is the so-called Kaluza-Klein reduction. This method is applicable if the higher-dimensional modes that are a solution of the equations of motion factorize into a product of four-dimensional modes and modes depending only on the higher-dimensional coordinates. This decomposition is possible in most higherdimensional models, which are typically constructed on a manifold M B 4 × k where M4 is a 4-dimensional manifold representing our four-dimensional spacetime and Bk is a k-dimensional compact manifold. The ansatz for the metric GMN of M B consistent with this setup is 4 × k

GMN = b diag[gµν (x), gmn(y)] , (1.8) where b is a constant, gµν is the metric of M4, and gmn is the metric of Bk. It is straightforward to verify that the d’Alembert operator 24+k decomposes as

24+k = 2k + 24 . (1.9) 16 1 Effective methods in braneworld dynamics

Therefore, most common Lagrangians, in particular that of the linearized gravitational field, will have mode-solutions which factorize into a four-dimensional and a k-dimensional part. Using this factorization, each (4 + k)-dimensional mode can be expanded in an infinite complete set of harmonics of the k-dimensional manifold. Each higher-dimensional mode is represented by an infinite number of modes. The mode belonging to the eigenvalue 0 of 2k is called zero mode whereas the modes to non-zero eigenvalues are called KK modes. The four-dimensional effective masses of the KK modes will have a contribution proportional to the eigenvalue of the corresponding harmonic. A four-dimensional effective action is obtained from the KK decomposition by inserting the mode sum into the action and integrating out the k higher dimensions. If the first non-zero eigenvalue is large, the first KK mode is very heavy. In this case it is often sufficient to truncate the action after the zero mode and ignore the dynamics of the KK-modes. This procedure is commonly known as KK reduction. Although the metric of the RS two-brane model (1.5) does not belong to the type (1.8) — the model is therefore, in a slight abuse of terminology, commonly said to have a non-factorizable geometry [7, 29] — a harmonic expansion of its linearized degrees of freedom is nevertheless possible.8 The wave operator of the RS setup in the coordinates of (1.5) reads d d2 4 2 F , 2 = + , (1.10) dy dy2 − l2 a2( y ) | | where 2 2 is the d’Alembert operator of four-dimensional Minkowski space and ≡ 4 we have introduced a( y ) exp( y /l). Taking into account the orbifold symmetry, | | ≡ −| | the junction conditions at the branes, i. e. at y = 0 and at y = d, are, d 2 + h = 0 . (1.11) dy l µν The harmonics solving the wave-equation for the y-direction, d F , m2 u(y) = 0 , (1.12) dy i with respect to the boundary conditions at the branes (1.11) are u (y) = N a2( y ) , (1.13) 0 0 | | u (y) = N [Y (lm )J (lm /a( y )) J (lm )Y (lm /a( y ))] , (1.14) i i 1 i 2 i | | − 1 i 2 i | | where N0, Ni are normalization constants. The Yn and Jn denote the Bessel and Neumann functions of nth order, respectively. We describe the calculations of the harmonics in more detail in Sec. 2.6. The KK masses, 2 = mi, are the roots of the eigenvalue equation for the harmonics (1.14), Y l√2/a J l√2 Y l√2 J l√2/a = 0 , (1.15) 1 1 − 1 1 where we have defined a a(d) = exp( d/l ). For a 1 Eq. (1.15) will be well ≡ − approximated by J1 l√2/a = 0 . (1.16)

8The presentation of the KK description of the RS model follows [66]. 1.3 Effective actions for warped braneworlds 17

The normalization for the harmonics is conveniently chosen as

d 2 1 dy a ( y ) ui(y)uj(y) = δij , (1.17) d | | l Z− where ui, uj also comprise the zero mode u0. Then one finds 1 1 N0 = , Ni = . (1.18) 2 √1 a a 2u2(lm ) u2(lm /a) − − i i − i i q We illustrate the use of the expansion in harmonics by sketching the derivation of linearized dynamics in the gravitational sector of the RS two-brane model following [68, 69]. In a specific gauge the most general perturbation of the five-dimensional metric is given by

ds2 = a2( y )(η + h )dxµdxν + (1 + φ(x))dy2 , (1.19) | | µν µν where hµν and φ(x) denote the perturbations.e The corresponding Lagrangian, quadratic in the fluctuations, is not diagonal in hµν and φ(x) but contains mixing terms e between the two perturbations. However, it is possible to express the field hµν by a combination of a new field hµν and φ soe that the Lagrangian in diagonalizes hµν and φ after using some residual gauge transformations. With the helpeof the eigenfunctions (1.13) and (1.14) we can decompose hµν as

∞ i hµν (x, y) = hµν (x)ui(y) . (1.20) Xi=0 Making an additional reparameterization of the ﬁeld φ, which transforms the kinetic term for the ﬁeld to the canonical form,

3d2 ϕ = φ , (1.21) l(a 2 1) s − − one obtains the action

2 l(a 1) 4 ∞ i 2 µν 2 i µν 2 Seﬀ = − dx hµν 4hi (mi/a) hµν hi + 2ϕ 4ϕ + Smat . − 64πG5 M − Z 4 i=0 h i X (1.22) Here, Smat denotes the action which describes the coupling of the linearized gravitational modes to matter on the branes,

1 4 µν 1 4 µν Smat = d x g(0) hµν (x, 0) T+ + d x g(d) hµν (x, d) T , (1.23) 2 Σ+ 2 Σ− − Z p Z p + where Tµν and Tµν− are the stress-energy tensors of matter on the Σ+- and Σ -brane, respectively. Decomposing the metric perturbation according to (1.20), one −ﬁnds for the interaction term on the Σ+-brane

1 4 0 µν ∞ i µν a + Smat,+ = d x hµν T+ + ui(0)hµν T+ ϕT , (1.24) 2 Σ− " − √3 # Z Xi=1 18 1 Effective methods in braneworld dynamics and for the interaction part of the action on the Σ -brane −

1 4 2 0 µν ∞ i µν 1 Smat, = d x a hµν T + ui(d)hµν T ϕT − , (1.25) − 2 Σ− " − − − √3a # Z Xi=1 µν where T = gµν T . For the lightest KK modes ui(0) and ui(d) are well approximated by u (0) a/J [lm /a] , u (d) a . (1.26) i ≈ 2 i i ≈ − An observer on the brane does his measurements with respect to a local Min- kowski frame. Thus, in order to infer observables on a brane one has to rescale its metric to Minkowski form. The induced metric on the Σ+-brane was chosen to be Minkowskian right from the beginning so the couplings and the KK-masses measured on the Σ+-brane can be directly read oﬀ from (1.22) and (1.24). The rescaling of the action for an observer on the Σ -brane will be deferred until we turn to phenomenological considerations in Sec. 3.3− . The KK description is most convenient if one is interested in the particle phenomenology of the RS model. With the explicit decomposition of the ﬁve-dimensional modes into the KK tower one can easily calculate cross sections and other high- energy physics observables relevant for collider physics [66, 67]. However, the KK decomposition has serious drawbacks in astrophysical applications. Already the determination of the potential of a static gravitational source requires the summation of the KK tower, and for cosmological considerations this method becomes even more involved.

1.3.2 Holographic description We now turn to another important description of the dynamics of the RS models which arises from a modification of the AdS/CFT correspondence. The approach uses the fact that the RS model shares the Anti-de Sitter background with the gravity-side of the AdS/CFT duality. We first discuss the duality between the bulk metric of the AdS space and the stress-energy tensor of the CFT. The AdS geometry is generated by a stack of branes in the near horizon limit [i. e. at y in the → ∞ coordinates of Eq. (1.5)], which support the CFT. The transition to the braneworld setup will then be made by truncating the spacetime by an end-of-the-world brane at a finite position in the bulk. This automatically yields an effective action for the on-brane dynamics in the RS one-brane model. The procedure described in the following was first presented in [24]. The interpretation of the RS two-brane model in terms of the AdS/CFT correspondence is has been discussed only little in the literature and, therefore, we will only give some short comments on this topic in Sec. 1.3.2.2. We will however return to the holographic interpretation of the RS two-brane model in Sec. 3.4.2 where we will generalize the holographic action to the two-brane model.

1.3.2.1 One-brane eﬀective gravity from AdS/CFT The AdS/CFT correspondence [55] relates the generating functional of a CFT on a d-dimensional boundary Md of AdSd+1 space to the partition function of string 1.3 Effective actions for warped braneworlds 19

theory on AdSd+1 by the equality

(0) (0) ZCFT[φ ] = Zstring[φ ] , (1.27)

(0) where φ are the boundary values for the string fields at Md and at the same time act as the sources for the CFT.9 In the correspondence the CFT is realized by N D3- branes in type IIB string theory, or by M5-branes in M-theory at the horizon of the AdS space. Reliable calculations of Zstring are only possible for the limit of an infinite number of branes, N . This corresponds to a very heavy and therefore, from the → ∞ gravitational point of view, virtually classical stack of branes generating a classical background spacetime. Thus, in the limit of large N the string partition function is well approximated by the tree-level functional of the corresponding supergravity action, Z [φ(0)] Ztree [φ(0)] = exp( S[φcl(φ(0))]) , (1.28) string ≈ string − where [φcl(φ(0))] denotes the field configuration which fulfills the classical equations of motion subject to the boundary conditions φ(0). If we restrict our attention to the bulk metric GMN , neglecting other fields φ, then it will be sufficient to take into account the stress-energy tensor of the conformal field theory, T CFT, because it is the only operator coupling to the metric. For the background-geometries under consideration the metric GMN does not induce a unique metric on the boundary Md because the metric will have a pole of second order at the boundary. However, GMN does determine an equivalence class of metrics, a conformal metric [g(0)µν ], on Md. In order to obtain a representative of [g(0)µν ] one chooses a function r in the bulk having a simple zero at the boundary 2 Md and restricts r GMN to Md. Different choices of r will result in different, but conformally equivalent, induced metrics on Md. Naively this means that the stress- energy tensor of the CFT, T CFT, will decouple from the metric, i. e. from gravitation. Considering the effective action of the CFT (i. e. the generating functional of the connected graphs), W [g ] = ln Z [g ], we then obtain a restatement CFT (0)µν − CFT (0)µν of the AdS/CFT correspondence,

WCFT[g(0)µν ] = Sgrav[GMN ] = Sbulk + Sboundary 1 1 1 = dd+1x√G (R 2Λ) + ddx√g K + α . (1.29) 16πG − 8πG ind 2 d+1 Z d+1 ZM4 Here R denotes the (d + 1)-dimensional Ricci scalar, Λ is the bulk cosmological constant, gindµν is the induced metric at the boundary Md and K the extrinsic curvature scalar on Md. The so-called Gibbons-Hawking boundary term guarantees the dependence of the action on ﬁrst derivatives of the metric only.10 The boundary contribution α was ﬁrst added to the action in [77] as a volume term on the boundary, i. e. as a surface tension, in analogy to the bulk cosmological constant. When we move

9Since we want to apply the AdS/CFT correspondence to the braneworld context by using the results of [74], our description of the features of the AdS/CFT duality which are relevant in our context follow Ref. [74]. 10Actually the necessity of this surface term to avoid second derivatives of the metric in the action was already discussed in [76]. 20 1 Effective methods in braneworld dynamics

the boundary Md to a ﬁnite position in the bulk and consider it as an end-of-the- world brane we will include in α not only the brane tension σ but also the Lagrangian density, Lmat, of matter ﬁelds, φ, on the brane,

16πG5 α = σ + Lmat(gind, φ) , (1.30) √gind For convenience we use a definition of the brane tension different from that in Sec. 1.2. The brane tension, according to the convention used in the present section, is related to that in Sec. 1.2 by multiplication with 8πG5. The brane tensions for the RS setup are then given by σ = 6/l. The action (1.29) is not well-defined for a metric GMN that fulfills the equations of motion because GMN has a pole of second order at the boundary Md. If one regularizes the action by a specific choice of g(0)µν this results in an explicit breaking of the conformal invariance of the CFT by the conformal anomaly. The conformal anomaly is an ultraviolet effect on the field theory side of the correspondence, but is induced by an infrared divergence on the gravitational side. The regularization procedure was first described in [74] which our presentation will follow closely. For the sake of simplicity we will restrict our consideration to a four-dimensional boundary M4. First we pick a specific metric g(0)µν on M4 from the conformal equivalence class [g(0)µν ]. According to [78] there is a distinguished coordinate system XM = (ρ, xµ) in the bulk in which the line-element of the metric GMN takes the form l2 ds2 = G dXM dXN = g (x, r)dxµdxν + dr2 , (1.31) MN r2 µν where gµν (x, r) is given by an expansion ofthe form 2 4 4 2 6 gµν (x, r) = g(0)µν + r g(2)µν + r g(4)µν + r ln(r )h(4)µν + O(r ) . (1.32) In this expression the subscript numbers do not only indicate the order of a term in the expansion in r but also the number of derivatives, with respect to xµ, it comprises. Obviously, for r 0 the induced metric g tends to g . In case of four → µν (0)µν dimensions the coefficients g(2)µν , g(4)µν and h(4)µν can be determined in dependence of g(0)µν by iteratively solving the bulk Einstein equations. The coefficients g(0)µν and g(2)µν are four-dimensional covariant tensors. The tensor g(2)µν is given by 1 1 g = R R g , (1.33) (2)µν 2 (0)µν − 6 (0) (0)µν where R(0)µν and R(0) are the Ricci tensor and Ricci scalar, respectively, constructed from the metric g(0)µν . The iteration procedure for g(x, r) breaks down for terms of higher-order in r in the sense that the coefficients g(i)µν , i 4, will no longer be co- µν≥ variant. Although g(4)µν is not a tensor its contraction, g(0)g(4)µν , is again covariant. Also h(4)µν is not covariant but its contraction with g(0)µν vanishes identically. The expansion (1.32) is discussed for arbitrary dimensions in [74, 79]. Since the expansion (1.32) contains only even powers of r it is convenient to conduct a change of variables, ρ r2. Then the line element and the expansion of ≡ the metric read 1 1 ds2 = G dXM dXN = l2 g (x, ρ)dxµdxν + dρ2 (1.34) MN ρ µν 4ρ2 1.3 Effective actions for warped braneworlds 21 and 2 3 g(x, ρ)µν = g(0)µν + ρg(2)µν + ρ ln ρ h(4)µν + O(ρ ) . (1.35) The regularization of (1.29) is achieved by restricting the bulk integral to a region ρ > where > 0 is an arbitrary cutoff, and then evaluating the integral for the five-dimensional metric GMN , which is a solution to the bulk Einstein equations. We can reformulate the action (1.29) in terms of the induced metric gµν on the boundary by expressing the bulk metric by Eq. (1.34),

S = Sbulk + Sboundary

1 4 ∞ √g 1 8 4 = d4x dρ + + ρ∂ + α √g . (1.36) 16πG l ρ3 ρ2 − l l ρ 5 ZM4 Z ρ=!

Substituting the expansion (1.35) for gµν (x, ρ) into Eq. (1.36) we arrive at

1 l4 6 l5α S = d4x g α + gµν g 16πG (0) 2 − l 2 (0) (2)µν 5 ZM4 " p l3 1 + ln Rµν R R2 + terms, finite for 0 . (1.37) 8 (0) (0)µν − 3 (0) → # In order to obtain the gravitational action for the AdS/CFT correspondence one would now have to construct counter terms to cancel the divergent terms of the action (1.37) and then take the limit 0. This procedure is carried out in [74]. → However, in order to make connection with the braneworld picture we consider the cutoff as a physical truncation: determines the position of a brane which cuts off the horizon region of the AdS bulk at the position [24, 36]. For this interpretation it is convenient to rewrite the action (1.37) in terms of the induced metric at the 2 cutoff, γµν = (l /)gµν on the brane at the position ρ = . This leads to the action

1 6 l l3 1 S = d4x√γ α + + ln µν 2 + . . . , (1.38) 16πG − l 2R 8 R Rµν − 3R 5 ZM4 where the dots indicate higher-derivative terms and the curvatures, , , are Rµν R constructed with respect to γµν . Since the cutoff position has dropped out most terms of Eq. (1.38), the action can no longer be regarded as an expansion in the small parameter . However, from the structure of the terms one realizes that the expansion can now be interpreted as a derivate expansion in the induced metric γµν. Since the terms denoted by dots in (1.38) will in general not be four-dimensional tensors [cf. the remark after Eq. (1.35)] the expansion will only be justified under the assumption that all higher derivatives of the metric are small on the scale of the AdS radius l, representing the characteristic curvature scale of the bulk. As explained above, the coefficient α comprises the sum of the brane tension and the Lagrangian of matter trapped on the brane, cf. Eq. (1.30). Substituting the RS value of the brane tension, σ = 6/l, we arrive at the action

l l2 1 S = d4x√γ + ln µν 2 + . . . + S , (1.39) RS 32πG R 4 R Rµν − 3R mat 5 ZM4 22 1 Effective methods in braneworld dynamics

4 where Smat = d x Lmat(γ, φ) is the action of the on-brane matter fields. The leading term in the derivative expansion (1.39) is just the usual Einstein- R Hilbert action with an effective gravitational constant 2G5/l. Note that in the present calculation we have not assumed a Z2 orbifold symmetry but restricted us to a single patch of AdS-space. Considering a setup with an orbifold symmetry, as in RS models, one would work along the same lines by conducting the calculation for two identical patches of AdS. This would lead to an effective gravitational constant G5/l on the brane which is the standard RS value found e. g. in the linearized analyses of [8, 21, 23]. On the CFT side of the correspondence the Z2 symmetry requires to consider two identical CFT’s — one for each AdS patch [64, 24]. The presence of the higher derivative terms in the braneworld action (1.39) can be interpreted in terms of the AdS/CFT correspondence as a coupling of the CFT to gravity on the brane, leading to the identification

l l2 1 W = d4x√γ ln µν 2 + . . . , (1.40) CFT 32πG 8 R Rµν − 3R 5 ZM4 where the dots again indicate the higher derivative terms [24]. Up to now we have neglected a subtlety which limits the actual utility of (1.39) for braneworld calculation beyond the scope of the recovery of the Einstein action at second order in the expansion in powers of the curvature. The scale appearing in the logarithm of the curvature-squared term’s prefactor is a length determining the distance of the brane from the coordinate singularity at ρ = 0 and thus a dimensionful quantity. Therefore the proper argument of the logarithm should be /s where s is a renormalization scale of dimension length (cf. [36]). Unfortunately, this renormalization scale cannot be determined intrinsically. Hence the strength of the higher-order curvature contributions to the action remains undetermined. We will recover the eﬀective action (1.39) in Sec. 3.4 by nonlocal methods which also yields the renormalization scale s. It will be identiﬁed, roughly speaking, with the energy scale of the on-brane modes under consideration, s = 1/√2.

1.3.2.2 The AdS/CFT interpretation of the two-brane model For the RS two-brane model no holographic action has yet been constructed. One can, however, make some qualitative statements on the role of the negative tension brane in a four-dimensional CFT interpretation: we have found that the gravitational infrared cutoﬀ induced by the positive tension brane in the RS model represents an ultraviolet breaking of the CFT. The negative tension brane resides in a regime where the on-brane gravitational interaction is strong. This will result in a breaking of the conformal invariance in the infrared regime of the CFT. The nature of this symmetry-breaking has been investigated in [80]. There it is argued that the immersion of the negative-tension brane into the bulk corresponds to spontaneous breaking of the conformal symmetry of the CFT. In [80] it is claimed that the expression for the generation functional of the CFT, (1.40), remains valid in the presence of the truncation of the bulk by the negative-tension brane because the calculations of [74] (cf. the previous section) are not modiﬁed due to the insertion of the second brane. This would imply that the stress-energy tensor of the CTF remains unaltered, which would indicate a symmetry breaking of spontaneous type. 1.3 Effective actions for warped braneworlds 23

However, the argument of [80] seems doubtful since the truncation of the bulk by the second brane should generate a surface term in the gravitational action which is not conformally invariant. This surface term should contribute to the stress-energy tensor of the CFT and therefore would invalidate the argument of [80]. Also [81] discusses the holographic interpretation of RS two-brane model. There, no explicit statement is made about the type of symmetry breaking by the negative-tension brane but all analogs presented there feature soft symmetry breaking. In Sec. 3.4.2 we calculate the surface term for the second brane as an extension of the method described in the previous section and thereby demonstrate that the symmetry breaking to be of soft type. The holographic action to be derived in Sec. 3.4.2 will be found to be identical with the nonlocal action for small brane separation to be derived in Sec. 2.8.1.

The nonlocal braneworld action

2.1 Introduction

We will now turn to our main task, the construction of the braneworld effective action (1.7) for the two-brane Randall-Sundrum model as a functional of two induced metrics gµν (x) and radion fields ψ(x) on the branes Σ . We obtain it perturbatively on the background of the RS solution up to quadratic order in brane curvatures Rµν and radions. Current interest in this effective action can be explained by attempts to devise cosmological scenarios utilizing the dynamics of either colliding [38, 40, 82, 83] or diverging [45] branes. Another motivation for the two-field braneworld action arises from the studies [84] and, especially [54], occupying a somewhat intermediate position between the Kaluza-Klein setting and the setting of (1.7). The authors of [84, 54] assume the two brane metrics g (x) to be conformally equivalent, g+ (x) g (x), and differ only µν µν ∼ µν− by the conformal (warp) factors at the branes. Thus, the braneworld action of [54] depends on one metric and two radion fields associated with the respective warp factors. This restriction of the total configuration space of the model suppresses important degrees of freedom which are in the focus of this study. The organization of the chapter is as follows. In Sec. 2.2.1 we explain the method of calculation for the nonlocal braneworld effective action. In Sec. 2.3 we present the final result for the two-field action in the two-brane RS model which we advocate here. This action is obtained as a quadratic form in Ricci curvatures and radion fields on the two branes. It is covariant with respect to two independent diffeomorphisms associated with these branes. The Secs. 2.4 and 2.5 are devoted to the derivation of the action. We begin with the construction of the effective equations of motion for the two-brane RS mode in, Sec. 2.4. Then we recover the action which generates these equations by a variational procedure in Sec. 2.5. The nonlocal form factors of the braneworld action are explicitly constructed in Sec. 2.6. In this section we also consider them in the lowest order of the derivative (2) expansion and show how and to what extent the four-dimensional Einstein theory is recovered on the branes. In Sec. 2.6.2 we present the form factors in the few lowest orders of the derivative (2) expansion. This expansion will be valid as long as the eigenvalues of 2, i. e. the energies of the four-dimensional modes are small. Then, in Sec. 2.7 we analyze the particle contents of the obtained action and show

25 26 2 The nonlocal braneworld action in Sec. 2.7.1 that the roots of the form factors generate the tower of Kaluza-Klein modes. In Sec. 2.7.2 and 2.7.3 we address problems of the particle interpretation arising in the scalar sector of the model and on the negative-tension brane for large brane separation. Moreover, we consider the limit of large interbrane distance1 which turns out to be the high-energy limit on the negative-tension brane and discuss the properties of the relevant nonlocal operators. To circumvent the problems encoun- tered in Secs. 2.7.2 and 2.7.3, we consider in Sec. 2.8 the reduced (one-field) effective action corresponding to the on-shell reduction in the sector of fields on the negative- tension brane. In this way we confirm the result of [45], where the braneworld action took the form of a Brans-Dicke type theory of the radion field non-minimally coupled to curvature. In the limit of large brane separation we confirm the realization of the AdS/CFT-correspondence principle by recovering the RS one-brane model. We also discuss applying Hartle boundary conditions at the AdS horizon and the problem of analytic continuation to the Euclidean spacetime. For the sake of clarity the nonlocal action for the RS one-brane model is also derived directly in the one-brane setting in Sec. 2.9.

2.2 The eﬀective action of brane-localized ﬁelds and the methods of its calculation

2.2.1 The structure of the braneworld effective action In the definition (1.7) the effective action by construction depends on the four- dimensional fields associated with the brane(s). The number of these fields equals the number of branes; each field is carried by one of the branes in the system. In the generalized RS setup, the braneworld effective action is generated by a path integral of the type (1.7),

DG exp iS[ G, g, φ ] = exp iSeff [ g, φ ] , (2.1) Z 4G(Σ) = g where the integration over bulk metrics runs subject to fixed induced metrics on the branes — the arguments of Seff [ g, φ ]. Here S[ G, g, φ ] is the action of the five- dimensional gravitational field with the metric GAB(x, y), A = (µ, 5), µ = 0, 1, 2, 3, propagating in the bulk spacetime (xA = (x, y), x = xµ, x5 = y). Matter fields φ are confined to the branes ΣI — four-dimensional timelike surfaces embedded in the bulk,

4 1/2 1 S[ G, g, φ ] = S5[ G ] + d x Lm(φ, ∂φ, g) g σ + [K] , (2.2) ΣI − 8πG5 XI Z 1 5 1/2 5 S5[ G ] = d x G R(G) 2Λ5 . (2.3) 16πG5 − Z5 M

The branes are enumerated by the index I and carry induced metrics g = gµν (x) and matter ﬁeld Lagrangians Lm(φ, ∂φ, g). We use a collective notation g, φ and σ for

1For the sake of clarity we denote the distance between the two branes as interbrane distance whereas the separation between two points on the same brane is called on-brane distance. 2.2 The effective action of brane-localized fields 27 the induced metrics, matter fields and tensions on all branes Σ. The bulk part of the action contains the five-dimensional gravitational and cosmological constants, G5 and Λ5, while the brane parts have four-dimensional cosmological constants σI . The Einstein-Hilbert bulk action (2.3) is accompanied by the brane ‘Gibbons-Hawking’ terms containing the jump of the extrinsic curvature trace [K] associated with both sides of each brane [62]2. In the tree-level approximation the path integral (2.1) is dominated by the stationary point of the action (2.2). Its variation is given as a sum of five- and four- dimensional integrals,

1 1 δS[ G, g, φ ] = d4x dy G1/2 5RAB 5R GAB + Λ GAB δG (x, y) −16πG − 2 5 AB 5 Z 4 1/2 1 µν µν 1 µν µν + d x g [K g K] + (T g σ) δgµν (x), (2.4) ΣI −16πG5 − 2 − XI Z where [Kµν gµν K] denotes the jump of the extrinsic curvature terms across the − brane, and T µν(x) is the corresponding four-dimensional stress-energy tensor of matter ﬁelds on the branes,

2 δS [g, φ] T µν (x) = m , (2.5) 1/2 g δgµν (x) 4 Sm[ g, φ ] = d x Lm(φ, ∂φ, g) . (2.6) ΣI XI Z The action is stationary if the integrands of both integrals in (2.4) vanish, which gives rise to Einstein equations in the bulk,

δS[ G, g, φ ] 1 1 G1/2 5RAB GAB 5R + Λ GAB = 0, (2.7) δG (x, y) ≡ −16πG − 2 5 AB 5 which are subject to (generalized) Neumann type boundary conditions — the well- known Israel junction conditions —

δS[ G, g, φ ] 1 1 g1/2 [Kµν gµν K] + g1/2(T µν gµν σ) = 0, (2.8) δgµν (x) ≡ −16πG5 − 2 − or to Dirichlet type boundary conditions corresponding to ﬁxed (induced) metrics on the branes, with δgµν = 0 in the variation (2.4),

4 Gµν = gµν (x) . (2.9) Σ

The solution of the Dirichlet problem is obviously a functional of the brane metrics, GAB = GAB[ gµν (x) ], and it enters the tree-level approximation for the path

2 The extrinsic curvature Kµν is defined as a projection of the tensor AnB on the brane (the ∇ vector nB is the outward unit normal, i.e. the normal pointing from the bulk to the brane). With this definition the normals on the two sides of the brane are oppositely oriented and the extrinsic curvature jump [Kµν ] actually equals the sum of the so-defined curvatures on both sides of the brane. 28 2 The nonlocal braneworld action

integral (2.1). In this approximation Seﬀ [ g, φ ] reduces to the original action (2.2)– (2.3) calculated on the background of the solution GAB[gµν (x)], of this Dirichlet boundary-value problem

Seﬀ [ g, φ ] = S[ G[ g ], g, φ ] + O(~) . (2.10)

Note that, with this deﬁnition, the matter part of the eﬀective action coincides with that of the brane action in (2.6)

Seﬀ [g, φ] = S4[ g ] + Sm[ g, φ], (2.11)

while all non-trivial dependence on g arising from functional integration is contained in S4[ g ]. Generically, the calculation of this quantity is available only by semiclassical expansion in the bulk in powers of ~. Again, in the tree-level approximation,

4 1/2 ~ S4[ g ] = S5[ G[g] ] d x gI σI + O( ). (2.12) − ΣI XI Z The Dirichlet problem (2.7), (2.9) can be regarded as an intermediate stage in solving the problem (2.7), (2.8). Indeed, given the action (2.11) from the solution of the Dirichlet problem (2.7), (2.9), one can further apply the variational procedure with respect to the induced metric gµν to get the eﬀective equations

δS [ g, φ ] δS [ g ] 1 eﬀ = 4 + g1/2T µν(x) = 0, (2.13) δgµν (x) δgµν (x) 2

which are equivalent to the Israel junction conditions — a part of the full system of the bulk-brane equations of motion (2.7), (2.8) (cf. the appendix of [85]). The procedure of solving this system of equations is split into two stages. First we solve it in the bulk subject to Dirichlet boundary conditions on the branes and substitute the result into the bulk action to get the off-shell brane effective action. In the second stage the variation of the brane effective action with respect to the four-dimensional metric yields the remaining set of equations to be solved. This observation suggests two equivalent methods of recovering the braneworld effective action. One method is straightforward — the direct substitution of the solution GAB = GAB[ gµν (x) ] of the Dirichlet problem (2.7), (2.9) into the five- dimensional action. The other method, which we choose to pursue in the following, is less direct, but technically is simpler: we will recover the effective action from the effective equations (2.13). Their left-hand side is considered as a variational derivative with respect to the brane metric(s) gµν and can be functionally integrated to yield Seff [ g, φ]. Therefore we will firstly obtain these effective equations by solving the bulk part of the equations of motion and by explicitly rewriting the Israel junction conditions in terms of gµν . The crucial point in the subsequent functional integration is the determination of a correct integrating factor. This will be based on the fact that the stress tensor of matter fields always enters the variational derivative of the effective action with the coefficient (1/2)g1/2 of Eq. (2.13). The two abovementioned methods have been shown to be equivalent in [86]. 2.2 The effective action of brane-localized fields 29

2.2.2 The role of radion fields As we have seen, the braneworld effective action induced from the bulk action has as its arguments only the brane metrics and brane matter fields. In such a setting, one does not at all meet the variables describing the embedding of the brane into a bulk. From the perspective of the five-dimensional equations of motion this can easily be understood because the dynamical equations of the embedding variables are automatically enforced in virtue of the bulk five-dimensional Einstein equations, Israel junction conditions (equivalent, as we have just mentioned, to the stationarity of the action with respect to the induced metric) and the matter equations of motion on the brane3. Another way to look at this property is to borrow an old idea from canonical gravity — the fact that three-geometry carries information about time [87]. In the braneworld context, this idea implies that the brane four-geometry carries the information about the location of the brane in the bulk. Indeed, the functional (2.10) can be viewed as the Hamilton-Jacobi functional of the boundary geometry. The only distinction from the canonical theory is that the role of the boundary is played by a timelike brane, rather than a spacelike hypersurface, and the role of time is played by the fifth (spacelike) coordinate determining the location of the brane (the conventional use of the Hamilton-Jacobi equation in cosmology in the long- wavelength limit [88] was recently extended to the braneworld context in several papers, see e. g. [73, 89]). Nevertheless, the usual treatment of braneworlds features a scalar degree of freedom describing a perturbation of the brane position in the bulk [90, 21, 22, 23, 37]. Therefore it is worth understanding the role of this scalar, usually called radion, from the viewpoint of the effective action. A possible way to recover the radion mode is to consider the structure of the effective action S4[ g ] introduced above. Already in the tree-level approximation this is a very complicated nonlocal functional of the metric. At most, we know it in the low-energy approximation when the curvature of the brane is small compared to the curvature of the AdS bulk. The coefficients of the corresponding expansion in powers of the curvature are nonlocal. This nonlocality can be of two different types. One type is expressible in terms of the massive Green function of the operator (2 M 2) and, therefore, in the low-derivative limit − is reducible to an infinite sequence of quasi-local terms like 1 1 2 n d4x √g R R d4x √g R R. (2.14) 2 M 2 ∼ M 2 M 2 n Z − Z X In the low-energy limit, all these terms are suppressed by inverse powers of the mass parameter given by the bulk curvature scale, M 2 Λ , 2/M 2 1, and ∼ 5 therefore comprise small short-distance corrections. Another type of nonlocality, in which there is no mass gap parameter,

4 1 4 1 µν d x √g R 2 R, d x √g Rµν 2 R , (2.15) Z Z is much stronger in the infrared regime and cannot be neglected. Counting the number of degrees of freedom in an essentially nonlocal field theory is rather subtle. 3In view of the five-dimensional diffeomorphism invariance of the full action its variational derivative with respect to the embedding variables can be linearly expressed in terms of the variational derivatives with respect to other fields, which explains this property. 30 2 The nonlocal braneworld action

However, sometimes the nonlocal action can be localized in terms of extra ﬁelds, which makes the local treatment of the theory possible. For instance, the ﬁrst of the nonlocal actions above can be reformulated as

S¯[g, ψ] d4x √g ( ψR + ψ2ψ ) (2.16) ∼ Z containing an extra scalar field ψ. The variational equation for ψ, δS¯/δψ = 0, yields ψ (1/2)R as a solution, which, when substituted into S¯[g, ψ], renders the original ∼ nonlocal structure of (2.15)4. Thus, we suggest that the scalar radion mode in the braneworld effective action can be recovered by a similar mechanism — localization of essential nonlocal expressions in the curvature in terms of the radion field. The fact that on-shell this field can be expressed nonlocally in terms of curvature matches with the known calculations of [37], where the radion mode was given in terms of the conformal part of the metric perturbation5. A similar mechanism in the two- dimensional context is the localization of the trace anomaly-generated Polyakov action [91] in terms of the conformal factor of the metric.

2.3 Two-brane Randall-Sundrum model: the ﬁnal answer for the two-ﬁeld braneworld action

The action of the two-brane RS model [7] is given by Eq. (2.2) where the index I = denotes two branes with tensions σ . The fifth dimension has the topology of a circle labeled by the coordinate y, d < y d, with an orbifold Z -identification − ≤ 2 of points y and y. The branes are located at antipodal fixed points of the orbifold, − y = y , y+ = 0, y = d. When the branes are empty, Lm(φ, ∂φ, gµν ) = 0, and their | −| tensions are opposite in sign and fine-tuned to the values of Λ5 and G5, 6 3 Λ5 = 2 , σ+ = σ = . (2.17) −l − − 4πG5l This model admits a solution with an AdS metric in the bulk (l is its curvature radius),

2 2 2 y /l µ ν ds = dy + e− | | ηµν dx dx , (2.18)

0 = y+ y y = d, and with a flat induced metric ηµν on both branes [7]. With ≤ | | ≤ − the fine tuning (2.17) this solution exists for arbitrary brane separation d — two flat

4The second of the structures (2.15) seems to require an additional symmetric tensor field for its µν localization. Interestingly, within the curvature expansion the nonlocal Lagrangian Rµν (1/2)R generates a variational derivative differing from that of R(1/2)R by a term proportional to the Ein- stein tensor (see Eq. (2.158) below), such that the difference between these two nonlocal Lagrangians can be simulated by the Einstein-Hilbert one. 5The nonlocal expression for ψ in terms of curvature and nonlocalities in the action requires boundary conditions which depend on the particular problem one is solving on the four-dimensional braneworld. For the scattering problem they are of Feynman chronological nature. For the Cauchy problem the non-localities imply retardation. In the no-boundary prescription of the cosmological state they can be derived by analytic continuation from the Euclidean section of the braneworld geometry [36, 37] (see Sec. 2.8.2 below). In what follows we shall not specify them explicitly. 2.3 The final answer for the two-field action 31 branes stay in equilibrium. Their flatness is the result of compensation between the bulk cosmological constant and brane tensions. Now consider the RS model with weak matter sources for metric perturbations hAB(x, y) on the background of the solution (2.18) [7, 21, 23, 22], 2 2 2 y /l µ ν A B ds = dy + e− | | ηµνdx dx + hAB(x, y) dx dx , (2.19) such that this five-dimensional metric induces two four-dimensional metrics on the branes according to 2 gµν (x) = a ηµν + hµν (x). (2.20) Here the scale factors a = a(y ) can be expressed in terms of the interbrane distance 2d/l a+ = 1, a = e− a , (2.21) − ≡ and hµν (x) are the perturbations by which the brane metrics gµν (x) differ from the (conformally) flat metrics of the RS solution (2.18)6. Our main result is the braneworld effective action (2.12) calculated for the boundary conditions (2.9) and the metric of the perturbed form (2.20). We calculate it in the approximation quadratic in perturbations, so that it represents a quadratic form in terms of the two-dimensional columns of fields hµν (x), h+ (x) h = µν . (2.22) µν h (x) µν− It should be emphasized that, in contrast to [54] and other studies of two-brane + scenarios, the metric perturbations hµν (x) and hµν− (x) are independent, which results in a much richer configuration space of the theory and leads to additional degrees of freedom responsible for interbrane interaction. The braneworld effective action is invariant under four-dimensional diffeomorphisms acting independently on each brane. In the linear approximation they reduce to transformations of the metric perturbations,

h h + f + f (2.23) µν → µν µ , ν ν , µ with two independent local vector field parameters fµ = fµ(x). Therefore, rather than in terms of the metric perturbations themselves, the action can be expanded in terms of the tensorial invariants of these transformations — the linearized Ricci tensors of hµν = hµν (x), 1 R = 2h + hλ + hλ h , (2.24) µν 2 − µν ν,λµ µ,λν − ,µν on the flat four-dimensional backgrounds of both branes7. Commas denote partial derivatives, raising and lowering of braneworld indices here and everywhere is performed with the flat four-dimensional metric ηµν , hλ ηλσh , h ηµν h , R = ηµν R , (2.25) ν ≡ σν ≡ µν µν 6 It is needless to emphasize that h (x) = hµν (x, y) because the induced brane metrics are µν 6 non-trivially related to (2.19) via brane embedding functions. 7 − − Strictly speaking, Rµν is the linearized Ricci tensor of the artificial metric ηµν + hµν . It differs 2 − − 2 from the linearized Ricci curvature of the second brane, Rµν (a η + h ) = Rµν /a , by a factor of a2. 32 2 The nonlocal braneworld action and 2 denotes the flat spacetime d’Alembertian

µν 2 = η ∂µ∂ν . (2.26)

Finally, we have to specify the variables which determine the embedding of the branes in the bulk. Due to metric perturbations the branes no longer stay at fixed values of the fifth coordinate. Up to four-dimensional diffeomorphisms (2.23), their embedding variables consist of two four-dimensional scalar fields, the radions ψ(x). According to the mechanism discussed in the previous section, the braneworld action can depend on these scalars. Their geometrically invariant meaning is re- vealed in a special coordinate system where the bulk metric perturbations hAB(x, y) of Eq. (2.19) satisfy the so called Randall-Sundrum gauge conditions, hA5 = 0, , ν µ hµν = hµ = 0. In this coordinate system the brane embeddings are given by the equations l Σ : y = y + ψ(x), y+ = 0, y = d. (2.27) a2 − In the approximation linear in perturbation fields and vector gauge parameters (in this approximation all these quantities are of the same order of magnitude, h (x) ψ (x) f (x)), the radion fields are invariant under the action of the µν ∼ ∼ µ diffeomorphisms (2.23). The braneworld effective action, which we advocate here, and which will be derived in the following two sections, is given in terms of the invariant fields of the above type, (R (x), ψ (x)), by the following spacetime integral of a quadratic 2 2 µν × form,

2 2 2 1 4 T 2F( ) µν 1 T K( ) 6F( ) S [ g , ψ] = d x R R + R − R 4 µν 16πG µν l222 6 l222 4 Z 1 T K(2) 1 3 2Ψ+ R 2Ψ+ R . (2.28) − 6 l222 6 It is to be supplemented by a standard action for matter ﬁelds on the branes if these are non-empty. In (2.28) G4 is an eﬀective four-dimensional gravitational coupling constant, G G = 5 , (2.29) 4 l µν T T R , Ψ and Rµν , Ψ are the two-dimensional columns

R+ (x) ψ+(x) R = µν , Ψ = (2.30) µν R (x) ψ (x) µν− − and rows

T + T + Rµν = Rµν(x) Rµν− (x) , Ψ = ψ (x) ψ−(x) , (2.31) h i h i of two sets of curvature perturbations and radion fields, associated with the two branes (T denotes the matrix and vector transposition). The indices are raised as above by the flat spacetime metric η and R ηµν R . The kernels of the µν ≡ µν 2.3 The final answer for the two-field action 33 quadratic forms in (2.28) are nonlocal operators — non-polynomial functions of the flat-spacetime d’Alembertian (2.26). They are both given in terms of the basic 2 2 × matrix-valued operator

F++(2) F+ (2) F(2) = − (2.32) F +(2) F (2) − −− and powers of the d’Alembertian8. In particular, the operator K(2) reads

1 0 K(2) = 2 F(2) + − l22. (2.33) 0 1/a2 The fundamental operator F(2) is the inverse of the operator-valued matrix G(2),

F(2)G(2) = I, (2.34) which is determined by the Green function of the following ﬁve-dimensional diﬀer- ential operator with Neumann boundary conditions,

2 2 d 4 (4) + G(x, y x0, y0) = δ (x, x0) δ(y y0), (2.35) dy2 − l2 a2(y) | − d 2 + G(x, y x0, y0) = 0. (2.36) dy l | y=y

The kernel of this Green function rewritten in x-space as the operator function of 2 parametrically depends on y and y0,

G(x, y x0, y0) = l G(y, y0 2) δ(x, x0). (2.37) | | Then, the dimensionless elements of the matrix G(2) = G (2), I, J = , in (2.34) IJ are G (2) = G(y , y 2). (2.38) IJ I J | In the next two sections we derive the braneworld effective action (2.28) and then consider its interpretation and applications in braneworld physics. In particular, the concrete form of the fundamental operator F(2) will be given in Sec. 2.6–2.8 where various energy limits of this nonlocal form factor will be considered in detail. Before this, however, let us make a few more comments on the structure of (2.28). Eq. (2.28) gives the braneworld effective action in the approximation quadratic in 3 curvature perturbations and radion fields ε = (Rµν , Ψ). All higher order terms O(ε ) are discarded. In this approximation it is sufficient to keep the nonlocal form factors of the quadratic form in (2.28) as flat-space ones and construct the d’Alembertian in Cartesian coordinates. In particular, in these coordinates there is no need to write explicitly the covariant density weights of g1/2(x) which become non-trivial

8Nonlocalities in form factors of the effective action require the prescription of boundary conditions which depend on the type of the physical problem one is solving (see footnote at the end of Sec. 2.2.1). In what follows we will assume that they are specified by a particular type of analytic continuation from the Euclidean spacetime in which these boundary conditions are trivial — Dirichlet boundary conditions at the Euclidean braneworld infinity, x . This continuation will | | → ∞ be discussed below in Sec. 2.8.2. 34 2 The nonlocal braneworld action only in higher orders of curvature expansion. In an arbitrary curvilinear coordinate system the expression (2.28) should be appropriately covariantized — partial derivatives replaced by flat-space covariant ones, g1/2(x)-weights included, etc. — by the technique of the covariant perturbation theory [92, 93]. In particular, the nonlocal operations like

4 + 4 4 + d x ϕ (x) F+ (2) ϕ−(x) = d x d y ϕ (x) F+ (x, y) ϕ−(y) (2.39) − − Z ZΣ+ ZΣ−

(ϕ(x) = Rµν (x), ψ(x)) should be understood as double integrals with two-point kernels

F+ (2) δ(x, y) = F+ (x, y) (2.40) − − including the required density factors and operators of five-dimensional parallel 9 transport from the point x of Σ+-brane to the point y on the other Σ -brane . − In the Cartesian coordinates and in this approximation all these subtleties are not very important. In expressions like (2.39) the integration domains Σ can be identified with one braneworld four-volume Σ, and ϕ(x) can be interpreted as two fields inhabiting this single spacetime.

2.4 The eﬀective equations of motion

As it was discussed above, we will recover the braneworld effective action from effective equations for two fields — the induced metrics on the branes, gµν (x). The effective equations as variations of the action with respect to brane metrics, see (2.13), are equivalent to Israel junction conditions with the five-dimensional metric coefficients GAB(x, y) (entering extrinsic curvatures of the branes) expressed in µν terms of gµν (x) and T (x). In order to disentangle the effective four-dimensional equations we have to solve the bulk equations of motion for GAB(x, y) in terms of µν gµν (x) and T (x) and substitute the result in the junction conditions. Implicitly, this has been done at the exact level in [52] where all unknown terms were isolated in form of the five-dimensional Weyl tensor and its derivatives. For the linearized theory this can be carried out explicitly along the lines of [21]. We start by linearizing the five-dimensional Einstein equations (2.7) in terms of metric perturbations hAB(x, y) on the background of the RS solution (2.19). These equations are invariant with respect to infinitesimal five-dimensional diffeomorphisms and, therefore, require a gauge. By applying these gauge transformations and using the five-dimensional Einstein equations, one can always go to the coordinate system in which only transverse-traceless hµν -components of the metric perturbations are non-vanishing10 [22, 23],

ν µ h55 = hµ5 = 0 , hµ , ν = 0 , h µ = 0 . (2.41)

9In this respect the covariantization procedure here is more complicated than in [92], where the nonlocal operations had to be taken within one spacetime with a single metric ﬁeld. 10Usually this is called the RS gauge. However, this is a combination of the gauge-ﬁxing procedure and the use of the non-dynamical (constraint) part of linearized Einstein equations in the bulk. 2.4 The effective equations of motion 35

In this gauge the linearized equations in the bulk simplify to a single second order equation, valid for each component of hµν ,

F (d/dy, 2) hµν (x, y) = 0 , (2.42) d2 4 2 F (d/dy, 2) + , (2.43) ≡ dy2 − l2 a2(y) where F (d/dy, 2) is just the five-dimensional d’Alembert operator on AdS space. The RS coordinate system is, however, not Gaußian normal relative to Σ — both branes in these coordinates are not located at constant values of the fifth coordinate y . Therefore, let us introduce two coordinate systems which are Gaußian normal relative to their respective branes and mark them by ( ) [22]. Metric perturbations with respect to these coordinate systems, denoted correspondingly by ( ) hµν (x, y), also have only µν-components, Σ+ is located at the constant value y+ in the (+)-coordinates, while Σ has in these coordinates a non-trivial embedding. Vice − versa, in ( )-coordinates Σ is located at the fixed value y , while Σ+ is embedded − − ( ) − non-trivially. Metric perturbations hµν (x, y) in ( )-coordinates are related to those of the RS coordinate system, hµν (x, y), by remnant coordinate transformations that respect the conditions h5A(x, y) = 0. Every such transformation is parameterized by one four-dimensional scalar field ξ(x) and one four-dimensional vector field ξµ(x) and has a fixed dependence on the fifth coordinate [21]. These transformations to the two Gaußian normal coordinate systems read

( ) 2 2 2 h (x, y) = h (x, y) + lξ (x) + η a (y) ξ(x) + a (y) ξ (x) . (2.44) µν µν , µν l µν (µ,ν)

Thus, they give rise to two radion fields ξ(x) and two four-dimensional vector fields ξµ(x). ( ) In the ( ) Gaußian normal coordinates, the five-dimensional metrics gµν (x, y) ( ) are given by Eq. (2.19) with perturbations hµν (x, y), 2 2 2 y /l µ ν ( ) µ ν ds = dy + e− | | ηµν dx dx + hµν (x, y) dx dx . (2.45) The extrinsic curvatures of Σ in these coordinates simplify to ( ) 1 dgµν (x, y) Kµν = , (2.46) ∓2 dy y=y where the sign originates from different orientations of the outward normals to the corresponding branes. Therefore, the linearized boundary conditions (2.8) take the form d 2 ( ) 1 + hµν (x, y) = 8πG5 Tµν ηµνT , (2.47) dy l y=y ∓ − 3 where11

4 αβ δSm Tµν a ηµα ηνβ T = 2ηµα ηνβ , (2.48) ≡ δhαβ µν T η T . (2.49) ≡ µν 11 In the deﬁnition of Tµν we have to deviate from the general rule of lowering the indices with the ﬂat metric ηµν because in the exact theory, the covariant stress tensor is related to the contravariant 2 one via the full metric gµν = aηµν + O(hµν ). 36 2 The nonlocal braneworld action

Here we took into account that due to the orbifold Z2-symmetry the curvature jump is [Kµν ] = 2Kµν . Substituting (2.44) in (2.47) and taking into account that (d/dy + 2/l)a2(y) = 0 we ﬁnally ﬁnd

d 2 1 + hµν (x, y) = 8πG5 Tµν ηµν T 2ξ,µν. (2.50) dy l y=y ∓ − 3 −

These are the linear boundary conditions on perturbations hµν (x, y) in the RS gauge12. By taking the trace of this equation one ﬁnds on account of the tracelessness of hµν the dynamical four-dimensional equation for radion ﬁelds [21],

8πG5 2ξ(x) = T (x). (2.51) 6

Due to the transversality of hµν and the four-dimensional conservation law for Tµν the divergence of (2.50) results in a diﬀerentiated version of Eq. (2.51) and, therefore, does not lead to new equations. Now the system of equations and the boundary conditions for the bulk metric perturbations and the radions is complete. Eq. (2.51) determines the radions, while the boundary value problem, (2.42) and (2.50), determines the perturbations in the bulk. The solution to this problem can be given in terms of the Green function of the operator (2.43) with homogeneous Neumann boundary conditions on the branes. This Green function satisﬁes (2.35)–(2.36). The desired solution reads

+ hµν (x, y) = dx0 G(x, y x0, y+) wµν (x0) dx0 G(x, y x0, y ) wµν− (x0), (2.52) | − | − Z Z 1 w = 8πG T η T 2ξ, , µν ∓ 5 µν − 3 µν − µν where we have denoted the right-hand sides of the (inhomogeneous) boundary conditions (2.50) by wµν . If we introduce the shorthand notation for the kernel of the ﬁve-dimensional Green function (2.37), then the x-integration signs can be omitted according to the simple rule

dx0 G(x, y x0, y0) w(x0) = l G(y, y0 2) w(x) (2.53) | | Z for any function w(x). The normalization factor l renders G(y, y 2) dimension- 0| less. This will be convenient, because we shall have to work with essential nonlocal operators expressible as nonlinear functions of the dimensionless argument l√2. We are interested in the effective dynamics of the observable fields only — the induced metric perturbations on the branes hµν (x). In the notation of Sec. 2.3 they form the column (2.22), and the Green function of Eq. (2.52) can be regarded as the 2 2-matrix × G++(2) G+ (2) G(2) = − (2.54) G +(2) G (2) − −− 12Strictly speaking, in the RS coordinate system the branes are shifted from the constant values y. However, this displacement is of first order of magnitude in ξ (x) h (x) and, therefore, it ∼ µν contributes to (2.50) only in higher orders of perturbation theory. 2.4 The effective equations of motion 37 with elements G (2) = G(y , y 2), I, J = (see Eq. (2.38)). With this notation IJ I J | the observable metric perturbations in the RS gauge read as the following (nonlocal) linear combination of stress tensors and radion fields,

+ 1 + + hµν (x, y+) 2 Tµν 3 ηµν T 2 ξ ,µν = 8πG5l G( ) − 1 2l G( ) . (2.55) hµν (x, y ) − Tµν− ηµν T − − ξ−,µν − − 3 − These perturbations do not, however, coincide with those of the induced metrics on branes, hµν . The latter are given by the values of the metric coeﬃcients in two ( ) respective Gaußian normal coordinate systems, hµν (x) hµν (x, y ). Therefore, ≡ hµν (x, y ) is related to hµν (x) by Eq. (2.44). In view of this relation, the perturbation of induced metrics reads

+ + 1 + 2 + hµν 2 Tµν 3 ηµν T ξ = 8πG5l G( ) − 1 + ηµν 2 h− − T − ηµν T − l a ξ− µν µν − 3 + ξ+ 1 0 2 ξ,µν (µ,ν) + l 2G( ) + 2 , (2.56) 0 1 − ξ− a ξ− − − ,µν " (µ,ν) # where we took into account that a+ = 1 and a a, Eq. (2.21). − ≡ From the four-dimensional viewpoint, the last two terms in this expression can be gauged away. Consequently the induced metric perturbations (we denote them in the new gauge by Hµν ) read

+ + 1 + 2 + Hµν 2 Tµν 3 ηµν T ξ = 8πG5l G( ) − 1 + ηµν 2 . (2.57) H− − T − ηµν T − l a ξ− µν µν − 3

The gauge conditions for Hµν (x) are unspecified yet and differ from those of hµν (x) (also unknown ones), but they will be determined later when covariantizing the effective equations of motion in terms of four-dimensional curvatures. In Eq. (2.57) we have a typical five-dimensional combination of the stress tensor and its trace, T 1 η T . This can be rewritten as the four-dimensional combination µν − 3 µν T 1 η T plus the contribution of the trace 1 η T which can be expressed in terms µν − 2 µν 6 µν of ξ according to (2.51). Then Eq. (2.57) takes the form

+ + 1 + + Hµν 2 Tµν 2 ηµν T 2 2 ξ = 8πG5l G( ) − 1 + lηµν Gξ( ) , (2.58) H− − T − ηµν T − ξ− µν µν − 2 where the new matrix-valued operator Gξ(2) given in terms of the brane separation parameter a reads

1 0 2 1 0 G (2) G(2) − + . (2.59) ξ ≡ 0 1 l2 2 0 a2 As we will see below, such a rearrangement is very illuminating when recovering the effective four-dimensional Einstein theory in the low-energy approximation. The next step consists in the covariantization of these equations. Thus far they are written in terms of brane metric perturbations hµν = Hµν in a particular gauge corresponding to the omission of gauge transformation terms on the right-hand side 38 2 The nonlocal braneworld action of (2.56). It is always useful to have the dynamical equations in gauge independent form. This form can be easily attained by rewriting them in terms of the curvature. For this purpose we first determine explicitly the gauge conditions for the perturbations Hµν and then express these perturbations in terms of the linearized Ricci tensors of brane metrics. The gauge for Hµν can be found by going back to the original metric perturbations hµν (x, y ) subject to the RS conditions of transverse- tracelessness. However, it is much easier to recover these gauge conditions directly from the equations of motion (2.57). By applying the first-order differential operator of harmonic gauge conditions to Hµν , one finds in view of (2.57) that

ν + 1 + + Hµ , ν 2 H, µ 2 2 ξ, µ ν − 1 = l Gξ( ) . (2.60) H − H− − ξ− µ , ν − 2 , µ , µ Here we used the conservation law for matter stress tensors which ensures their complete cancellation in the above equation. Eq. (2.60) serves as a set of generalized harmonic gauge conditions on the brane metrics Hµν and the radions ξ. These gauges nontrivially intertwine the ﬁelds on two branes because of the generically non-diagonal form of the matrix Gξ(2). Now, one can use them in Eq. (2.24) for the linearized Ricci tensor, rewritten in the form 1 1 2R = 2h + (hλ h ) + (hλ h ) + O(h2), (2.61) µν − µν ν,λ − 2 ,ν , µ µ,λ − 2 ,µ , ν

in order to obtain the equation for Hµν that, in its turn, can be solved by iteration in powers of Rµν and ξ [92]. In the linear approximation the solution reads H+ 2 R+ ξ+ µν = µν 2l G (2) , µν , (2.62) H −2 R − ξ ξ µν− µν− ,−µν where we suppress all higher order terms in Rνν, ξ. Using (2.62) in Eq. (2.57) we rewrite the latter in terms of the linearized Einstein tensors E = R 1 η R of the brane metrics, µν µν − 2 µν + + + 2 Eµν Tµν ξ + 8πG5 l G(2) 2l ( µ ν ηµν 2) Gξ(2) = 0. −2 E− T − − ∇ ∇ − ξ− µν µν (2.63) These equations are covariant and equally valid for arbitrary perturbations hµν taken in any gauge, not necessarily coinciding with that of Hµν . As it was discussed earlier, we now have to find the action that generates these equations by a variational procedure. For this purpose we have to find the integrating factor — an overall matrix valued coefficient — by means of which the left-hand side of (2.63) enters the variational derivative of the action, see (2.13). To find it, we note that the matter fields living on the branes are directly coupled to four-dimensional metrics gµν only via their stress tensors,

+ 1/2 µν δ/δgµν 1 g+ T+ Sm[gµν , φ] = 1/2 µν . (2.64) δ/δgµν− 2 " g T # − − Moreover, the matter action additively enters the full eﬀective action (2.11). There- fore, in the linear approximation the overall coeﬃcients of T µν in the variational 2.5 The recovery of the effective action 39

1 4 derivatives of the action should be 2 a (remember that we work in Cartesian co- 1/2 4 µν ordinates, so that g = a + O(hµν )). To achieve this minimal coupling of T µν we raise the indices in (2.63) with η and apply the operator F(2)/16πG5l, where F(2) is the matrix-valued inverse of the Green function G(2), Eq. (2.34). Then, the left hand side of equation (2.63) takes the desired form of a variational derivative with respect of the metric

δ/δg+ 1 F(2) Eµν 1 g1/2T µν µν S [ g , ξ , φ ] = + + + + eﬀ 2 µν 1/2 µν δ/δgµν− −8πG5l E 2 " g T # − − − + 1 µ ν µν ξ ( η 2) F(2) Gξ(2) . (2.65) − 8πG ∇ ∇ − ξ− 5

µα νβ Here we take into account that in view of the deﬁnition (2.48) η η Tαβ = a4 T µν = g1/2 T µν . Our next goal is to rewrite the equations for the radions (2.51) also as variational equations of the braneworld eﬀective action. Again, matter cannot be directly coupled to radions as follows from the structure of matter actions additively entering the whole action. Therefore, the stress tensor part of (2.51) should be reexpressed in terms of the curvature. By taking the trace of the gravitational equations (2.65) and using (2.51) one can exclude the stress tensor traces in terms of the scalar curvatures R and radions ξ, so that the dynamical equations for the latter reduce to

2 R+ 1 0 ξ+ F(2) = 6l2 − F(2) Gξ(2) . (2.66) 2 R− 0 1 − ξ−

Making use of (2.59) they take the surprisingly simple form

+ + R 6 1 0 2 ξ = 2 . (2.67) R− − l 0 a ξ−

Eq. (2.67) has a direct physical interpretation. It is the linearized equation of motion for perturbations of the brane worldsheet coupled to bulk metric perturbations. In the next section we recover the braneworld eﬀective action that generates the full set of equations (2.65) and (2.67).

2.5 The recovery of the braneworld eﬀective action

From the structure of (2.65) it is obvious that the graviton-radion part of the full braneworld action S4 [ gµν , ξ] is given by the sum of the purely gravitational part, the non-minimal coupling of radions to curvatures and the radion action itself,

S4 [ gµν , ξ] = Sgrav[ gµν ] + Sn m[ gµν , ξ] + Srad [ gµν , ξ]. (2.68) − 40 2 The nonlocal braneworld action

13 Sgrav and Sn m give the corresponding contributions to the variational derivatives − + 2 µν δ/δgµν 1 F( ) E+ Sgrav = µν , (2.69) δ/δgµν− −8πG5l 2 E − + + δ/δgµν 1 µ ν µν ξ Sn m = ( η 2) F(2) Gξ(2) , (2.70) δ/δg− − −8πG ∇ ∇ − ξ− µν 5

while Srad[ gµν , ξ] still has to be found by comparing its radion variational derivative with (2.67). Due to the linearity of Eq. (2.69) in metric perturbations, Sgrav can be obtained from its right-hand side by contracting with the row of metric perturbations + [Hµν, Hµν− ] and integrating over four-dimensional brane spacetime. Using (2.62), one can convert the result to the ﬁnal form quadratic in Ricci curvatures, 2 µν 1 4 + F( ) E+ Sgrav[ g] = d x Rµν Rµν− 2 µν . (2.71) 8πG5l 2 E Z h i − In view of the variational equation for the Ricci scalar, δR = ( µ ν gµν 2)δg + ∇ ∇ − µν O(Rµν ), the non-minimal curvature coupling term in (2.70) is easily recovered as

+ 1 4 + ξ Sn m[ g, ξ] = d x R R− F(2)Gξ(2) . (2.72) − −8πG5 ξ− Z h i With this non-minimal coupling the equations of motion for the radions read

+ + + δ/δξ δ/δξ 1 T 2 T 2 R Srad + Sn m = Srad Gξ ( )F ( ) = 0, δ/δξ− − δ/δξ− − 8πG R− 5 (2.73) where T denotes the transposition of the corresponding 2 2 matrices. Then the × comparison with (2.67) shows that

+ + δ/δξ 6 1 0 2 1 0 ξ Srad = 3 2 K( ) 2 , (2.74) δ/δξ− −8πG l 0 a 0 a ξ− 5 1 0 K(2) GT (2)FT (2) l22 . (2.75) ≡ 0 1/a2 ξ

As it follows from the deﬁnition of Gξ(2), Eq. (2.59), the matrix K(2) is given by Eq. (2.33). Thus, it is symmetric in view of the symmetry of F(2). Equation (2.74) is therefore integrable and the radion action acquires the quadratic form with this new matrix valued operator K(2) as a kernel. The equations become simpler if we introduce instead of ξ the following new dimensionless radion ﬁelds ξ+ ψ+ = , l 2 ξ− ψ− = a , (2.76) l

13 These variational derivatives are calculated discarding higher order terms in ε = (Rµν , ξ) which we do not include explicitly. The same rule will be used in what follows for the braneworld eﬀective action calculated to second order in these ﬁelds inclusive. 2.6 Green functions 41 in terms of which the non-minimal coupling and radion actions read

2 + l 4 + K( ) ψ Sn m[ g, ξ] = d x R R− 2 , (2.77) − −8πG l 2 ψ− 5 Z h i + 3 4 + ψ Srad[ g, ξ] = d x ψ ψ− K(2) . (2.78) −8πG5l ψ− Z h i Thus, the sum of gravitational (2.71), non-minimal (2.77) and radion (2.78) parts form the full braneworld effective action in the approximation quadratic in fields. Collecting the radion and non-minimal parts one can easily disentangle the last term of (2.28) — the 2 2 quadratic form in (2Ψ+R/6) — plus an extra term quadratic × in the Ricci scalars on the branes R. When combined with (2.71), the latter gives rise to the first two terms of (2.28). This accomplishes the derivation of the basic result advocated in Sec. 2.3. The nonlocal kernels in the quadratic forms of (2.28) are built in terms of 2 2-matrix-valued operators F(2) and K(2) which we analyze × in the next section.

2.6 Green functions

To construct the operators F(2) and K(2) we need the Green function of the operator (2.43) — the solution of the boundary value problem (2.35)–(2.36). This problem simpliﬁes to the Bessel function equation in terms of the new variable z, y l z = l exp , a(y) = . (2.79) l z For the dimensionless function G¯(z, z 2) G(y, y 2) deﬁned by Eq. (2.37) it reads 0| ≡ 0| as d d 4 z + z2 G¯(z, z0 2) = δ(z z0), (2.80) dz dz − z | − d 2 z G¯(z, z0 2) = 0. (2.81) dz | z=z

This Green function can be built in terms of the basis functions u (z) = u (z 2) of | Eq. (2.80) — two linearly independent solutions of the homogeneous Bessel equation satisfying Neumann boundary conditions at z+ and z , respectively, − d d 4 z + z2 u (z) = 0, z+ z z , (2.82) dz dz − z ≤ ≤ − d z2u (z) = 0. (2.83) dz z=z

They are given by linear combinations of Bessel and Neumann functions of the second order, Z2(z√2) = (J2(z√2), Y2(z√2)), with the coeﬃcients easily derivable 2 2 from the boundary conditions on account of the relation (d/dx)x Z2(x) = x Z1(x), 2 2 u (z) = Y1J2(z√ ) J1Y2(z√ ), (2.84) − 2 2 J1 J1(z √ ), Y1 Y1(z √ ). (2.85) ≡ ≡ 42 2 The nonlocal braneworld action

In what follows we introduce the abbreviation for the Bessel function of any order 2 Zν Zν(z √ ) to avoid excessive use of their diﬀerent arguments. ≡ In terms of u (z), the Green function has a well known representation,

u (z)u+(z0) u+(z)u (z0) G¯(z, z0 2) = θ(z z0) − + θ(z0 z) − , (2.86) | − ∆ − ∆ where θ(z) is the step function, θ(z) = 1, z 0, θ(z) = 0, z < 0, and ∆ is the ≥ conserved Wronskian inner product of basis functions in the space of solutions of Eq. (2.82),

d d 2 + + ∆ z u+(z) u (z) u (z) u+(z) = J1 Y1− Y1 J1− . (2.87) ≡ dz − − − dz π − One can directly check that this two-point function satisfies the boundary value problem of the above type — it is continuous at z = z0 and has at this point a jump in the first-order derivative that generates the delta function on the right-hand side of (2.80). Actually, this is a typical Green function of the Sturm-Liouville problem on a finite segment, which in the physical context is analogous to the positive and negative frequency decomposition of the Feynman Green function on an infinite time axis. The calculation of the 2 2-matrix Green function (2.54) on the basis of (2.86) × requires the knowledge of u+(z ) and u (z ). Some of them simplify to elementary − functions, 2 u (z ) = Y1J2 J1Y2 = . (2.88) − πz √2 Using this result together with (2.87) in (2.86), one finds the following exact expression for G(2),

2 2 1 1 √ z u (z+) 2 − − π G( ) = + +  2  . (2.89) 2z+z J Y − J − Y 2 − 1 1 − 1 1 √ z+u+(z )  π −    The Green functions in G(2) have ﬁrst been calculated in [94]. Interestingly, its inverse F(2) can be represented in a form, in essence, dual to this expression. Indeed, for the inversion of the matrix (2.89) we need its determinant which can be shown to equal14

+ + 2 1 Y2−J2 Y2 J2− det G( ) = + − + , (2.90) −z+z 2 Y −J Y J − − 1 1 − 1 1 so that F(2) assumes a form structurally similar to (2.89)

2 2 1 √ z+u+(z ) 2 − −π F( ) = + +  2  . (2.91) −J Y − J − Y 2 2 2 − 2 2 √ z u (z+)  −π − −    14To derive this expression for the determinant one should explicitly use the ﬁrst of the expressions for u(z) from (2.88) in the oﬀ-diagonal elements of (2.89), and then notice that the numerator of the determinant factorizes into the product (Y −J + Y +J −)(J + Y − J − Y +). 2 2 − 2 2 1 1 − 1 1 2.6 Green functions 43

This exact expression for F(2) will be important for us in what follows when considering the low-energy approximation, because a direct expansion of (2.91) in powers of 2 turns out to be much simpler than the expansion of (2.89) with its subsequent inversion.

2.6.1 The low-energy limit — recovery of Einstein theory The Green function (2.89) and its inverse operator (2.91) are nonlinear functions of 2. This fact corresponds to the essentially nonlocal nature of the effective four- dimensional theory induced on the branes from the bulk. A remarkable property of the one-brane RS model is however that the low-energy approximation of the effective four-dimensional theory is (quasi)local and corresponds to Einstein gravity minimally coupled to matter fields on the branes. In the two–brane model the situation is more complicated — the low-energy theory belongs to the Brans-Dicke type with a non-minimal curvature coupling of the extra scalar field that mediates the interaction of the metric with the trace of the matter stress tensor [21]. We begin considering the low-energy effective theory in the leading order approximation by taking the limit 2 0 of the Green function G(2), (2.89). By using the → asymptotics of small argument for Bessel and Neumann functions, x x3 J (x) = + O(x5), (2.92) 1 2 − 16 x2 x4 J (x) = + O(x6), (2.93) 2 8 − 96 2 x x 1 5 Y (x) = + C + ln + x3 + O(x5), (2.94) 1 −πx π 2 − 2 32π 4 1 x2 x 3 Y (x) = + 2 C + ln + O(x4) , (2.95) 2 −πx2 − π 8π 2 − 2 h i one finds the low-energy behavior of various ingredients of Eqs. (2.89) and (2.91) as expansions in integer and half-integer powers of 2, 3 3 + + 1 x x+ x+x x 1 x+ x 4 J1 Y1− J1−Y1 = − + − ln − + − + O(x ) , − π x+ − x 2π x+ 8π x − x+ − − (2.96) 2 2 4 4 + + 1 x x+ 1 2 2 1 x+ x 4 J2 Y2− J2−Y2 = −2 2 + (x x+) + 2 −2 + O(x ) , − 2π x − x 8π − − 24π x − x + + − − (2.97) 2 3 2 x+ x+ 1 x 1 x+ 3 u+(z ) = 2 + − 2 + O(x ) , (2.98) − π x 2π − 4π x+ − 4π x − − 2 3 2 x x 1 x+ 1 x 3 − − u (z+) = 2 + 2− + O(x ) . (2.99) − π x+ 2π − 4π x − 4π x+ − Here, x = z √2, and O(xn) = O(2n/2) denotes terms of n-th power in √2. Then we have in the leading order approximation in 2 0 → 2 1 1 a2 G(2) = + O(l22), (2.100) l22 1 a2 a2 a4 − 44 2 The nonlocal braneworld action and 2 a2 1 1 G (2) = − + O(l22), (2.101) ξ −l22 1 a2 1 1 − − d/l where a is the brane separation parameter, a = e− . With these Green functions the effective equations (2.63) take the form of the following two (linearized) Einstein equations,

d/l + + + 1 e− + E = 8π G T + G−T − + ( η 2)(ξ ξ−) , (2.102) µν µν µν l sinh(d/l) ∇µ∇ν − µν − d/l 2d/l + + 1 e− + E− = 8πe− G T + G−T − + ( η 2)(ξ ξ−) . µν µν µν l sinh(d/l) ∇µ∇ν − µν − (2.103)

Matter sources on both branes are coupled to gravity with the eﬀective four-dimensional gravitational constants G [21] depending on the brane separation,

d/l G5 e G = , (2.104) l 2 sinh(d/l) and there is also a non-minimal curvature coupling to a particular combination of radion fields (ξ+ ξ ) which, obviously, describes dynamical disturbances of the − − interbrane distance. Thus, the low-energy theory reduces to the generalized Brans- Dicke model – the fact that was first observed in [21] (see also [95] on the realization of this property in braneworld scenarios with bulk scalar fields). For large distance between the branes, a = e d/l 1, the metric field on the − positive-tension brane decouples from all fields on the other brane and from radions because

+ G5 1 2d/l G = G4, G− G4e− 0, d , (2.105) l 1 e 2d/l → ' → → ∞ − − and the low-energy theory on this brane reduces to Einstein gravity with the four- dimensional gravitational constant (2.29). This is a manifestation of the so-called graviton zero-mode localization in the one-brane RS model [7] or the AdS/CFT- correspondence [24, 36, 37]. The recovery of Einstein theory on the positive tension brane is the result of a non-trivial cancellation between the contributions of the stress tensor trace and the radion in the right-hand side of the effective equations of motion (2.57). This cancellation leads to Eq. (2.58) with an exponentially small matrix G (2) a2 0, (2.101), of the non-minimal coupling to radions. ξ ∼ → For finite interbrane distance, both stress tensors Tµν contribute to the right- hand sides of Einstein’s equations (2.102)–(2.103). They contribute to Eµν with 2d/l different strengths (their contribution to the negative tension brane metric is e− + + times weaker), but in one and the same combination G Tµν + G−Tµν− . This, maybe physically obvious, fact has a crucial consequence for the structure of the off-shell extension of the effective braneworld theory. Mathematically this property manifests itself in the degeneracy of the leading-order matrix Green function (2.100). Another degeneration that occurs in the low-energy limit is the fact that among two radion fields, that were introduced on a kinematical ground as independent entities, only 2.6 Green functions 45 their combination (ξ+ ξ ) is dynamical. Apparently, this is the explanation why − − only one radion field is usually considered as a dynamically relevant variable in the two-brane RS model (see [54] where this property was explained by the homogeneity of the AdS background). Such a degeneration, as we see, is not fundamental, but turns out to be an artifact of the adopted low-energy approximation scheme. The braneworld effective action requires the knowledge of F(2) — the inverse of the Greens’s function G(2). The latter, as we have just seen, is degenerate in the lowest order of the 2-expansion (2.100), which is why we have to go beyond this approximation. In this expansion one can single out two distinctly different physical regimes characterized by the values of the dimensionless parameters l22 and a. One regime corresponds to small energies compared to the AdS-scale, 2 1/l2 or l22 1, and small or finite interbrane distance a = O(1), such that l22/a2 1. The last requirement implies that the physical energy range is small also at the negative tension brane, because the physical energy (or inverse physical distance) at µν this brane is determined by g ∂µ∂ν = √2/a. − Another regime correspondsq to small energies on the positive tension brane and large energies on the negative tension one, l22 1 and l22/a2 1, when a 1. This limit includes, in particular, the situation when the second brane is pushed to infinity of the fifth coordinate (the AdS horizon) and qualitatively is equivalent to the one-brane situation15. Below we will start with systematic studies of these two regimes of the low-energy expansion and its particle interpretation.

2.6.2 Low-energy derivative expansion

Here we consider the derivative expansion of the nonlocal form factors in the ﬁrst of the low-energy regimes, corresponding to small or ﬁnite interbrane distance. In this regime the arguments of both sets of Bessel and Neumann functions in (2.89)–(2.91) are small l√2 l√2 1, 1. (2.106) a

Therefore, one can expand G(2) to higher than zeroth order in 2 and, thus, make it explicitly invertible. Since G(2) is degenerate in the zeroth order, its matrix determinant is at least O(1/2) rather than O(1/22) and, therefore, one should expect that its inverse F(2) will be a massive operator — its expansion in powers of 2 will start with the mass matrix O(20). This degeneracy leads also to an additional diﬃculty — in order to achieve the kinetic term in F(2) linear in 2 one would have to calculate the Green function G(2) to O(22) inclusive. Fortunately, instead of inverting the Green function expansion, we have the exact expression (2.91) for F(2) that can be directly expanded to a needed order. Thus, using the small argument

15The situation with an infinitely remote second brane is not entirely equivalent to the one- brane model, because the Israel junction condition on the second brane is different from the usually assumed Hartle boundary conditions on the AdS horizon, see Sec. 2.8.2. 46 2 The nonlocal braneworld action expansions of Bessel functions in (2.91) we get F(2) = M + D 2 + F(2)l222 + O(23), (2.107) l2 − F F 1 4 a4 a2 M = − , (2.108) F l2 1 a4 a2 1 − − 1 a2 a2 + 3 2 D = − , (2.109) F 6(1 + a2)2 2 3 + a 2 − where the components of the matrix F(2) are rather lengthy (but we will need them below because they will qualitatively effect the low-energy behavior in the radion sector), (1 a2)3(3 + a2) ln a 4 + 5a2 4a4 + a6 F (2) = − − , ++ 72a2(1 + a2)3 − 4(1 a4)2 − 96a2(1 a4) 2 3 2 −4 2 − 4 6 (2) (1 a ) (3a + 1) a ln a 1 4a + 5a + 4a F = − 4 2 3 4 2 − 4 4 , −− 72a (1 + a ) − 4(1 a ) − 96a (1 a ) 2 3 2 − 2 4 − (2) (1 a ) a ln a 1 8a + a F+ = 2 − 2 3 + 4 2 −2 4 . (2.110) − 36a (1 + a ) 4(1 a ) − 96a (1 a ) − − The matrix coefficients of this 2-expansion are non-diagonal and, therefore, nontrivially entangle the fields on both branes. An important property of the lowest-order coefficient — the mass matrix MF — is that it is degenerate and has rank one. As we will see below, this fact will guarantee the presence of one massless graviton in the spectrum of the braneworld action. The low-energy expansion for K(2) follows from that of the operator F(2), (2.107)–(2.110), K(2) = M + D 2 + K(2) l222 + O(23) , (2.111) l2 − K K where 8 1 a4 a2 M = 2M = − , (2.112) K F l2 1 a4 a2 1 − − 2 4a2 2a4 1 a2 D = − − − (2.113) K 3(1 + a2)2 1 a2 4 + 2a 2 − − (2) (2) and K = 2F . Similarly to the graviton operator F(2), the mass matrix MK is degenerate, but in contrast to DF the kinetic matrix DK is indefinite, 4 det D = < 0, (2.114) K −(1 a2)2 − so that MK and DK are not even granted to be simultaneously diagonalizable. This follows, in particular, from the calculation of the matrix determinant of the radion operator 4 ln a det K(2) = (l22)2 + O[ (l22)3], (2.115) 1 a4 − which turns out to be at least quadratic in 2 and negative (remember that a < 1)16.

16The quadratic dependence of this determinant on 2 implies that the characteristic equation for 2.7 The particle content 47

2.7 The particle content of the two-ﬁeld braneworld action

The quadratic approximation for the action and its nonlocal formfactors obviously determines the spectrum of excitations in the theory. Here we show that in the graviton sector this spectrum corresponds to the tower of Kaluza-Klein modes well-known from a conventional KK setup. The graviton sector arises when one decomposes metric perturbations on both branes into irreducible components — transverse-traceless tensor, vector and scalar parts, , ν µν hµν = γµν + ϕηµν + fµ,ν + fν,µ , γµν = η γµν = 0. (2.116) On substituting this decomposition in the linearized curvatures of (2.28) one ﬁnds that the vector parts do not contribute to the action, and the latter reduces to the sum of the graviton and scalar sectors,

S4 [ gµν , ψ] = Sgraviton[γµν ] + Sscalar[ ϕ, ψ] . (2.117) The graviton part is entirely determined by the operator F(2) and reads 2 µν 1 4 1 + F( ) γ+ Sgraviton[γµν ] = d x [ γµν γµν− ] 2 µν , (2.118) 16πG4 2 l γ Z − while the scalar sector consists of the radion fields of Eq. (2.30) and the doublets of the trace (or conformal) parts of the metric perturbations ϕ, + ϕ (x) T + Φ = , Φ = ϕ (x) ϕ−(x) . (2.119) ϕ−(x) h i Their action reads as the following 4 4 quadratic form with the block-structure × kernel composed of 2 2 matrix-valued operators F(2) and K(2), × Sscalar[ ϕ, ψ] 3 F(2) K(2) Φ = d4x ΦT ΨT − . (2.120) 16πG l2 K(2) 2K(2) Ψ 4 Z − The direct particle interpretation of the quadratic actions (2.118) and (2.120) is still not obvious because their kernels are nonlocal and nontrivially entangle the fields on both branes and intertwine radions Ψ with conformal scalars Φ. Moreover, they both originate from a double diffeomorphism invariant and nonlocal theory with the covariant action (2.28). Therefore, the full set of fields should be carefully analyzed for the purpose of establishing their physical or purely gauge status, which is very non-trivial for nonlocal theories, if at all unambiguous17. Below we partly clarify these questions in the low-energy approximation when the theory gets (at least partially) localized. We begin with the simplest sector of the theory — the graviton contribution. simultaneous diagonalization of MK and DK (with 2 playing the role of the eigenvalue parameter) has only one multiple root equal to zero. 17This implies the application of the four-dimensional gauge-fixing procedure. Note that we have already used one — the five-dimensional gauge-fixing procedure in the bulk when solving the bulk equations of motion in order to obtain the effective four-dimensional theory with the remaining four- dimensional diffeomorphisms. The latter also have to be gauged away to reach a correct bookkeeping of the physical degrees of freedom in the effective braneworld theory. 48 2 The nonlocal braneworld action

2.7.1 The graviton sector Excitations in the graviton sector are the transverse-traceless basis functions v(x) = vµν (x) of the operator F(2), v+(x) F(2) v (x) = 0, v (x) = n . (2.121) n n v (x) n− The existence condition for zero-vectors of the 2 2 matrix operator F(2), × det F(2) = 0, (2.122)

serves as the equation for the masses of these propagating modes mn, (2 m2 ) v (x) = 0, (2.123) − n n 2 2 determined by the roots = mn of (2.122). In view of Eq. (2.90) these roots coincide with the zeros of the following combination of Bessel functions

+ + 2 Y −J Y J − = 0 . (2.124) 1 1 − 1 1 But the same equation (2.124) determines the spectrum of the KK modes — the zeros of the Wronskian of basis functions u (z), Eq. (2.87), entering the Neumann Green function (2.86). Therefore, a conventional tower of massive KK modes is contained in the spectrum of the braneworld effective action. We will discuss the particle interpretation of the transverse-traceless sector in extent in Sec. 3.2 where we establish the connection between our present approach and the KK description of the RS two-brane model. The derivative expansion of the previous section allows one to get a reliable 2 description for the massless sector of the spectrum, m0 = 0. In this sector Eq. (2.121) reduces to MF v0(x) = 0 and implies that v0(x) is a zero mode of the mass matrix MF — the property guaranteed by the degeneracy of this matrix, mentioned above. The non-diagonal nature of (2.108) implies that this mode is still the collective 2 + excitation of tensor perturbations v0(x) on both branes, but v0−(x) = a v0 (x) < v+(x) and for large interbrane separation, a 0, the negative brane component 0 → tends to zero, v−(x) 0 , so that the massless graviton is essentially localized on 0 → the positive-tension brane. This is, certainly, another manifestation of the recovery of Einstein theory on this brane for the two-brane RS model. The attempt to describe massive modes within the derivative expansion for F(2) turns out to be inconsistent. Indeed, the truncation of the series (2.107) on the second term, for the goal of finding the first massive level, leads to the equation det( M +D 2) = 0 instead of (2.122). It implies the simultaneous diagonalization − F F of both mass and kinetic term matrices in the basis of the first two (massless and massive) propagating modes (which is possible in view of the positive-definiteness of the kinetic matrix DF ). But, unfortunately, the massive root of this equation, 2 = M 2 24a2(1 + a2)/l2(1 a2)2 a2/l2, strongly violates the second of the ≡ − low-energy conditions (2.106). The excited massive graviton mode always turns out to be in the physical high-energy domain on the negative-tension brane, and the low-energy description fails. For this reason we restrict ourselves to the massless sector of the theory — the effects of massive modes will be considered in Sec. 3.2. 2.7 The particle content 49

2.7.2 Problems with the scalar sector of the theory The 4 4 matrix in the scalar sector of the model (2.120) can be reduced to a × block-diagonal form by the transformation

2 1 F(2) K(2) 2 0 K(2) 2F(2) 0 − = − , 0 1 K(2) 2K(2) 1 1 0 2K(2) − − (2.125) which further simpliﬁes, because in view of the relation (2.33), its upper-left block becomes the diagonal matrix proportional to 2. Thus it reads

l22 0 0 − 0 a 2l22 0 , (2.126)  −  0 0 2K(2) −   in which one of the trace modes turns out to be a massless ghost, and only the radion operator continues to intertwine nontrivially the ﬁelds on both branes. The action diagonalizes in terms of the conformal modes and the (redeﬁned) radion modes 2Ψ Φ, −

3 4 + + 1 S [ ϕ, ψ] = d x ϕ 2ϕ + ϕ−2ϕ− scalar 64πG − a2 4 Z 3 K(2) d4x 2Ψ Φ T 2Ψ Φ . (2.127) − 32πG − l2 − 4 Z In this parameterization that the sector of conformal modes is entirely local, and the last 2 2 radion quadratic form here corresponds to the last term of the action (2.28) × quadratic in the left-hand side of the radion equations of motion (2.67), 2Ψ+R/6 = 2(2Ψ Φ)/2. − Thus, apart from the difficulty of diagonalizing the low-energy action, the four- dimensional scalar sector of trace and radion modes contains two ghosts — one negative kinetic term in the trace sector and one in the radion sector. It is very tempting to conclude that the model is severely unstable, because in the ground state it does not have a well-defined spectrum of positive energy excitations. Equivalently, the Euclidean version of the braneworld action is not positive definite in view of the indefiniteness of the quadratic form (2.28). This conclusion is, however, premature for two reasons. One reason is related to the way the dependence on the radion fields was incorporated in the action (2.28). In Sec. 2.4 radions were introduced at the kinematical level as gauge variables relating the RS coordinate system to two Gaußian normal systems associated with the two branes. Then it was demanded that the kinematical relations between radions and stress tensor traces (2.51) should be generated as dynamical equations from the braneworld action (2.28). This has led to the last term in (2.28) quadratic in (2Ψ + R/6). Thus, this term might be regarded as the result of the off-shell extension in the radion sector. The on-shell reduction of (2.28) simply corresponds to the exclusion of the radions in terms of the metric fields or, equivalently, to omitting the last term in (2.28). Alternatively, one might consider the on-shell reduction in the sector of conformal modes Φ. Each of these truncations 50 2 The nonlocal braneworld action reduces the configuration space of the quadratic action (2.28) to a certain subspace on which the action can have alternative convexity properties. The on-shell reduction in the conformal sector — exclusion of Φ in terms of radions Ψ — corresponds to the replacement of the original action by the new one 2 3 4 T KΨ( ) S [ ϕ, ψ] S [ ψ] = d x Ψ Ψ (2.128) scalar ⇒ scalar 16πG l2 4 Z with the kernel of the scalar sector going over to the new kernel of the radion quadratic form,

F(2) K(2) 1 − KΨ(2) = 2K(2) + K(2) K(2). (2.129) K(2) 2K(2) ⇒ − F(2) − In view of (2.33) this kernel expresses in terms of the Green function G(2)

2 1 0 2 2 1 0 1 0 KΨ(2) = 2l 2 − + (l 2) − G(2) − 0 a 2 0 a 2 0 a 2 − − − (2.130) and has a low-energy expansion which begins with the kinetic term — the mass matrix is identically vanishing,

K (2) Ψ = D 2 + D(2) l222 + O(23), (2.131) l2 Ψ Ψ 2 a2 1 DΨ = 2 − 2 . (2.132) 1 a 1 a− − − The matrix of the kinetic term DΨ is, however, degenerate and has rank one. There- fore, the kinetic term of one of the modes in (2.128) is determined by the next order 2 (2) in — the matrix DΨ with components 1 ln a 3 a2 D(2) = + , Ψ ++ 1 a2 1 a2 4 − 4 − − (2) 1 ln a 1 1 DΨ + = 2 2 2 , − 1 a −1 a − 4a − 4 − − (2) 1 ln a 3 1 DΨ = 2 2 + 2 4 . (2.133) −− 1 a 1 a 4a − 4a − − (2) One can show that this matrix is positive definite, so that DΨ and DΨ are simultaneously diagonalizable and describe one low-energy massless particle and one massless dipole ghost — a particle with a Lagrangian of the form ϕ22ϕ. Therefore, the situation does not qualitatively improve as compared to the full off-shell scalar action. The alternative on-shell reduction in the radion sector corresponds to the exclusion of radion fields and the replacement of the original 4 4 operator by the × operator acting in the conformal sector,

F(2) K(2) 1 − K (2) = F(2) + K(2) K(2), (2.134) K(2) 2K(2) ⇒ Φ − 2K(2) − 2.7 The particle content 51 which on account of (2.33) turns out to be very simple, 2 2 K (2) = (l 2/2) diag [ 1, a− ] , (2.135) Φ − and leads to two massless local scalar fields, one of them being a ghost, 3 4 +2 + 1 2 Sscalar[ ϕ, ψ] Sscalar[ ϕ] = d x ϕ ϕ + 2 ϕ− ϕ− . (2.136) ⇒ 32πG4 − a Z This on-shell reduction is physically the most transparent one, and it again seems to result in physical instability. However, one should remember that the transition to irreducible components of fields does not guarantee that all of them are physical. For example, in pure Einstein theory the conformal mode survives in the action after the transition to the transverse-traceless decomposition (2.116) and, moreover, looks like a ghost because of the wrong sign of its kinetic term. But it does not signify any instability, because this mode is not dynamically independent in view of constraints — the non-dynamical part of equations of motion. Due to them this mode does not require independent initial conditions — they turn out to be defined by the initial conditions for the three-dimensional transverse-traceless metric components which form the physical sector18. Similarly to the case of local Einstein theory we have to analyze the constraint restrictions in braneworld theory. The starting point of this analysis should be the covariant action (2.28), because constraints cannot be grasped in terms of irreducible fields. This analysis is, however, aggravated by the fundamentally nonlocal nature of the effective four-dimensional theory, which survives even in the low-energy approximation. This nonlocality originates, in particular, from the massive gravitons in the spectrum of the theory, because mass terms in the graviton sector (2.123) in covari- 2 ant form correspond to structures like Rµν (1/2 )Rµν . The way to handle implicit constraints in a nonlocal theory apparently consists in the preliminary localization of its action in terms of extra four-dimensional fields. The anticipated solution of the stability problem might be based on the radion on-shell reduction (2.136). One field with the correct kinetic term remains dynamical, while the ghost scalar in (2.136) similarly to the conformal mode in Einstein theory gets ruled out by constraints and gauge transformations19.

2.7.3 Large interbrane distance In view of the discussion of the previous section, it is instructive to consider in the low-energy approximation on the positive-tension brane the case of large brane separation, when a 1 and l√2 l√2 1, 1. (2.137) a 18An old curse of the Euclidean gravity theory is that the conformal mode with its wrong sign of the kinetic term violates positivity of the Euclidean Einstein action. However, unlike in Lorentzian theory this mode cannot be ruled out by the Euclidean version of the constraints, because of the qualitatively different setting of the boundary value problem in Euclidean theory. Many efforts has been spent to resolve this difficulty, but no satisfactory solution has been attained thus far, see, however, [96]. The indefinite sign of the kinetic term in the Einstein theory is directly related to the attractive nature of gravity [97] and, thus, is physically important even despite the non-dynamical status of the conformal mode. 19We are grateful to D. V. Nesterov for this conjecture. 52 2 The nonlocal braneworld action

This range of coordinate distances 1/√2 corresponds to the long-distance approximation on the Σ -brane and to the physical short-distance limit, a/√2 l, on the + Σ -brane. Now one should use the asymptotic expressions of large arguments of the − Bessel functions (Jν−, Yν−), ν = 1, 2,

2a l√2 π πν J − cos , ν ' πl21/2 a − 4 − 2 r 2a l√2 π πν Y − sin , (2.138) ν ' πl21/2 a − 4 − 2 r + + and as before the small-argument expansions for (Jν , Yν ). Then, in the leading order the operator F(2) reads

l22 l22 2 2J −  2  F(2) . (2.139) ' 2  l 2 J − l   1 √2   a   2J2− −J2−    In contrast to the case of small brane separation, the short-distance corrections to this matrix operator contain a nonlocal 22 ln 2-term. Here we present it for the F++(2)-element,

l22 (l22)2 F (2) = + k (2) + O (l22)3 , (2.140) ++ 2 2 2 h i 1 4 Y2− k2(2) = ln 2C + π (2.141) 4 l22 − J 2−

(the meaning of the subscript in k2(2) will become clear in Sec. 2.8.2 below). This is a manifestation of the well-known phenomenon of AdS/CFT-correspondence [55, 24, 64, 98] when typical quantum field theoretical effects in four-dimensional theory can be generated from the classical theory in the bulk. The AdS/CFT-duality exists for the boundary theory in the AdS bulk when the brane, treated as a boundary of the AdS spacetime, tends to infinity. In the wording of the two-brane RS model [24, 45] this situation qualitatively corresponds to the case of the negative tension brane tending to the horizon of the AdS spacetime, y , or a 1. → ∞ The non-logarithmic terms of Eq. (2.141) include the functions with an infinite series of poles at l22 π2a2(n + 3/4)2, n = 0, 1, ..., contained in the ratio ' Y − J − l√2 π 2 1 tan . (2.142) J ' −J ' a − 4 2− 2− Interestingly, these poles arise in the wave operator of the theory, rather than only in its Green function. This is an artifact of the nonlocality, when both the propagator and its inverse are nonlocal. The poles are separated by the sequence of roots on the positive real axis of the 2-plane,

π2a2 1 π2a2 2 = m2 (n + 3/4)2 1 + (n + 3/4) + ... , (2.143) n ' l2 l2 2 2.8 The reduced effective action 53 which correspond to the tower of massive KK modes in the energy range (2.137)20. They are excited for large brane separation, a 1, because a small energy range at the positive tension brane turns out to be the high energy range at the second brane. The presence of both zeros and poles in the wave operator F(2) can, apparently, be explained by the duality relation between the Dirichlet and Neumann Green functions in braneworld physics [86]. According to this relation the Neumann Green function in the bulk when restricted to branes (which is exactly the deﬁnition of F(2)) is the inverse of the bulk Dirichlet Green function properly diﬀerentiated with respect to its two arguments and also restricted to branes. Therefore, the necessarily existing poles of the Dirichlet Green function generate zeros of the Neumann one and vice versa. The presence of zeros, each of which is located between a relevant pair of neighboring poles of F(2), is actually very important, because this guarantees the 1 positivity of residues of all poles in G(2) = F− (2) or the normal non-ghost nature of all massive modes.

2.8 The reduced eﬀective action

If we take the usual viewpoint of the braneworld framework, that our visible world is one of the branes embedded in a higher-dimensional bulk, then the fields living on other branes are not directly observable. In this case the effective dynamics should be formulated in terms of fields on the visible brane. In the two-brane RS model this is equivalent to constructing a reduced action — an action with on-shell reduction for the invisible fields in terms of the visible ones. This reduction is nothing but the tree- level procedure of tracing (or integrating) out the unobservable variables. It implies that in the two-brane action we have to exclude the fields on the invisible brane in terms of those on the visible one. As the latter we choose the positive-tension brane. One of the reasons for such a choice is that the low-energy dynamics on this brane is closest to four-dimensional Einstein theory, while that of the negative-tension brane is encumbered with the problems discussed above — intrusion into the high-energy domain, excitation of massive modes, etc. The reduced action is obtained by varying the two-field action consisting of (2.117) and the action for matter on the branes with respect to the fields on the brane considered invisible. The equations of motion are then formally solved for the fields on the hidden brane. Finally one inserts the expressions for the hidden fields into the two-field action. In this way one arrives at an action depending only on the metric, the radion and the matter fields on the visible brane. We perform the reduction of the action to the Σ+-fields separately in the graviton and scalar sectors. In the graviton sector (2.118) the on-shell reduction — the exclusion of γµν− - + perturbations in terms of γµν = gµν (in what follows we omit the label +, because only one field remains) — corresponds to the replacement of the original action by the new one, 2 red 1 4 + Fred( ) µν S [ γ ] S [ g ] = d x √gγ γ , (2.144) graviton µν ⇒ graviton µν 16πG µν 2 l2 + 4 Z 20In this range the above expressions for poles and zeros of F(2) are valid for large n satisfying 1 π(n + 3/4) 1/a. 54 2 The nonlocal braneworld action

with the original kernel F(2) going over to the new one-component kernel Fred(2) according to the following simple prescription 1 F(2) Fred(2) = F++(2) F+ (2) F +(2) . (2.145) ⇒ − − F (2) − −− It is useful to rewrite (2.144) back to the covariant form in terms of (linearized) Ricci curvatures on a single visible brane, 1 F (2) 1 F (2) Sred [g ] = d4x √g R red Rµν R red R . (2.146) graviton µν 8πG µν l2 22 − 3 l2 22 4 Z A similar reduction in the scalar sector (2.120),

Sscalar[ ϕ, ψ] 3 (2) ϕ+ Sred [ ϕ+, ψ+] = d4x ϕ+ ψ+ Kred , (2.147) ⇒ scalar 16πG l2 ψ+ 4 Z can be easily performed by rewriting the kernel of this form explicitly as a 4 4 + + × matrix acting on columns of conformal modes and radions (ϕ , ψ , ϕ−, ψ−), F K 1 K K − − 2 K 2K K 2K  − ++ − +  (2) = − , (2.148) K  1 K K F K   − 2 −   K 2K K 2K   − + −   − −−  where the -superscripts of each 2 2 block refer to all of its elements. The reduction × to (ϕ+, ψ+) leads, by the pattern of Eq. (2.145), to the matrix operator

F K (2) = − Kred K 2K − ++ 1 1 K K F K − 1 K K − 2 − − 2 , (2.149) − K 2K + K 2K K 2K + − − − −− − − which in view of Eq. (2.33) simpliﬁes to the form 1 1 l22 1 0 (2) = K (2) 2 − , (2.150) Kred − red 1 2 − 2 0 0 − where 1 det K(2) Kred(2) = K++(2) K+ (2) K +(2) = . (2.151) − − K (2) − K (2) −− −− Finally, we express the conformal mode in terms of the (linearized) Ricci scalar ϕ+ = (1/32)R, and denote the radion by ψ+ = ψ. Then the combination of (2.147) − together with the graviton part (2.146) yields the reduced action in its covariant form

1 2F 1 S [g , ψ] = d4x √g R red Rµν gµν R red µν 16πG µν l222 − 2 4 Z 1 1 2F R 2K R R red R 6l2 2ψ + red 2ψ + . (2.152) −6 2 − l222 − 6 l222 6 2.8 The reduced effective action 55

Here we have deliberately singled out the term bilinear in the Ricci and Einstein tensors, because this form will be useful in comparing the low-energy approximation of this action with the Einstein action. Below we analyze this result in the two low-energy regimes (2.106) and (2.137).

2.8.1 Small interbrane distance In the regime of small or ﬁnite brane separation (2.106), the calculation of the reduced operator (2.145) gives, on using (2.107)–(2.109), a very simple result,

l22 (l22)2 F (2) = (1 a2) + κ (a) + O[ (l22)3], (2.153) red 2 − 2 1 1 1 1 κ (a) = ln (1 a2) (1 a2)2 . (2.154) 1 4 a2 − − − 2 − The reduced operator turns out to be massless, and this is a corollary of the degenerate nature of the mass matrix (2.108), det MF = 0, because Fred(2) = det F(2)/F (2) = O(2). Similarly, in view of (2.115), the reduced operator in the −− radion sector (2.151) is at least quadratic in 2,

2 2 2 3 Kred(2) = κ2(a)(l 2) + O[ (l 2) ], (2.155) 1 1 κ (a) = ln , (2.156) 2 4 a2 so that the low-energy radion turns out to be a dipole ghost. µν After substituting (2.153) into (2.152), the first term bilinear in Rµν and R 1 µν − 2 g R seems to remain nonlocal. However, this term is nothing but the part of the local Einstein action, which is quadratic in metric perturbations δgµν = hµν on a flat spacetime background. To see this, note that, up to diffeomorphism with some vector field parameter fµ, this perturbation can be nonlocally rewritten in terms of the (linearized) Ricci tensor,

h = (2/2)R + f + f + O[ R2 ] . (2.157) µν − µν µ , ν ν , µ µν When substituted into the quadratic part of the Einstein action it takes an explicitly nonlocal form in terms of Ricci curvatures,

1 1 1 δ2 d4x √gR = d4x √g Rµν gµν R R + O[ R3 ], (2.158) 2 − 2 2 µν µν Z Z which is exactly the ﬁrst term of (2.152) with the ﬁrst-order in 2 approximation for Fred(2), (2.153). The part of the Einstein action linear in perturbations is a total divergence, which we disregard here, and the zeroth order term is identically vanishing. Therefore, this term can be rewritten as the local Einstein action linear in scalar curvature with the a-dependent four-dimensional gravitational constant (2.105),

G G (a) = 4 = G+, (2.159) 4 1 a2 − 56 2 The nonlocal braneworld action

where G4 is given by (2.29). The second term of (2.152) with Fred(2) given by (2.153) stays nonlocal although this nonlocality is suppressed by the factor a2 < 1. The quadratic in 2 contribution to (2.153) generates the local part of the action 2 2 quadratic in curvatures which enter in a special combination Rµν R /3. This 2 − combination diﬀers from the square of the Weyl tensor Cµναβ by the density of the Gauß-Bonnet invariant

E = R2 4R2 + R2 , (2.160) µναβ − µν R2 2 R2 = C2 E , (2.161) µν − 3 µναβ − which can be omitted under the integral sign (because it is topologically invariant and in the quadratic order in metric perturbations explicitly reduces to the total surface term [92, 93]). Thus, collecting diﬀerent contributions, we get the reduced braneworld action in the low-energy regime of ﬁnite interbrane distance

1 a2 1 S [ g , ψ ] = d4x √g (1 a2)R R R red µν 16πG − − 6 2 4 Z R 2 l2 6l2κ (a) 2ψ + + κ (a) C2 . (2.162) − 2 6 2 1 µναβ The first term here confirms the recovery of Einstein theory on the positive tension brane with the well-known expression for the effective gravitational constant G4(a) [21, 22]. The higher derivative nature of the radion does not really imply physical instability, because ψ can hardly be treated as non-gauge variable21. Its equation of motion, R 2 2ψ + = 0, (2.163) 6 implies that the on-shell restriction of (2.162) leaves us with the first two metric-field terms. In the low-energy regime the first three terms dominate over the local short- distance Weyl-squared part. The action consisting of the first three terms in (2.162) was already derived in [45] by a simplified (and, strictly speaking, not very legitimate) method — by just freezing to zero all field perturbations on the invisible brane. Here this derivation is justified within a consistent scheme accounting for the fact that, even without matter sources on Σ , the field perturbations on the visible − brane induce nontrivial fields on the invisible one, and they contribute to the full effective action. Interestingly, however, the result turns out to be the same as in [45]. The second term of (2.162) is nonlocal. However, according to the discussion in [45], it can be localized in terms of an extra scalar field. Actually, this extra field can

21Indeed, in Sec. 2.4 radions were introduced as gauge variables relating the RS coordinate system to two Gaußian systems associated with two branes. Then it was demanded that the kinematical relations between radions and stress tensor traces (2.51) should be generated as dynamical equations from the braneworld action (2.28). This has led to the last term in (2.28) quadratic in (2Ψ + R/6). Thus, this term can be regarded as the result of the off-shell extension in the radion sector. The on-shell reduction simply corresponds to the exclusion of radions in terms of the metric fields or, equivalently, to omitting the last term in (2.28). 2.8 The reduced effective action 57 be identified with the radion itself up to some nonlocal reparameterization. Indeed, the following reparameterization from ψ to the new field ϕ,

3 1 κ2(a) 2 1 ϕ = a 1 2R l 2 ψ + R (2.164) 4πG4 " − 6 − 6 # r r − converts the action (2.162) to the local form

1 1 1 S [ g , ϕ ] = d4x √g ϕ2 R + ϕ2ϕ red µν 16πG − 12 2 Z " 4 2 l 2 + κ1(a) Cµναβ (2.165) 32πG4 # whose first three (low-derivative) terms were derived in [45] by a simplified procedure. The field ϕ introduced here by the formal transformation (2.164) directly arose in [45] as a local redefinition of the radion field relating the RS coordinates to the Gaußian normal coordinates associated with the positive tension brane. It is non- minimally coupled to the curvature, and in [45] it was used to play the role of the inflaton generating inflation in the presence of a small detuning between the values of the brane tensions σ from their RS values (2.17). Initial conditions for inflation in [45] were suggested within the tunneling wavefunction scheme [99] modified according to the braneworld creation framework [84, 36, 37]. In this framework the Lorentzian spacetime arises as a result of analytic continuation from the Euclidean space describing the classically forbidden (underbarrier) state of the gravitational field. Note, in connection with this, that the transformation (2.164) is well-defined only in Euclidean spacetime with the negative-definite operator 2. Thus, the justification of the off-shell reparameterization between the actions (2.162) and (2.165) comes from the Euclidean version of the theory, which underlies the braneworld creation scheme22. In the next section we shall extend this justification even further by resorting to Hartle boundary conditions on the AdS horizon.

2.8.2 Large interbrane distance and Hartle boundary conditions In the limit a 0 the nonlocal and correspondingly non-minimal terms of (2.162) → and (2.165) vanish and the low-energy model seems to reproduce the Einstein theory. However, this limit corresponds to another energy regime (2.137) in which one should use the expressions (2.139)–(2.141) in order to obtain the reduced operator (2.145). Then the latter, up to quadratic in 2 terms inclusive, reads as

22 22 2 22 2 l (l ) 4 (l ) π Y2− a Fred(2) = + ln 2C + + . (2.166) 2 8 l22 − 2 4 J 2l√2J J 2− 1− 2− As in (2.140) it involves the logarithmic nonlocality (2.141) in 22-terms. Moreover, the last term here simpliﬁes to the ratio of the ﬁrst order Bessel functions Y1−/J1−, so

22The transformation (2.164) is complex-valued for timelike momenta, but its on-shell restriction in the radion sector is real. 58 2 The nonlocal braneworld action

that Fred(2) takes a form very similar to that of large interbrane separation (2.140),

22 22 2 2 l (l ) 4 Y1− 22 3 Fred( ) = + ln 2 2C + π + O (l ) . (2.167) 2 8 l 2 − J − 1 h i The calculation of the radion operator (2.151) with K(2) following from (2.139) for l22/a2 1 results in 2 2 Kred(2) = (l 2) k2(2), (2.168)

where k2(2) is deﬁned by (2.141). Thus, Fred(2) and Kred(2) are given by the following two nonlocal operators,

1 4 Yν− kν (2) = ln 2C + π , ν = 1, 2, (2.169) 4 l22 − J ν− and the reduced (one-brane) action ﬁnally reads

1 1 S [g , ψ] = d4x √g R + l2 R k (2)Rµν R k (2) R red µν 16πG µν 1 − 3 1 4 Z R R 6l2 2ψ + k (2) 2ψ + . (2.170) − 6 2 6 If one expresses the curvature part of the action in terms of the Weyl tensor it reads

1 l2 S [g , ψ] = d4x √g R + C k (2)Cµναβ red µν 16πG 2 µναβ 1 4 Z R R 6l2 2ψ + k (2) 2ψ + . (2.171) − 6 2 6 Here terms quadratic in curvature (which we again rewrote in terms of the Weyl squared combination in view of integration by parts) represent short distance corrections with form factors whose logarithmic parts have an interpretation in terms of the AdS/CFT-correspondence [55, 24, 64, 98, 45]. Their Bessel-function parts are more subtle for interpretation and less universal. When taken literally they give rise to the massive resonances discussed above in Sec. 2.7.3. However, with the usual Wick rotation prescription 2 2 + iε these ratios tend to → Y l π πν ν− tan √2 + iε i, a 0, (2.172) J ' a − 4 − 2 → → ν− and both form factors (2.169) for 2 < 0 (Euclidean or spacelike momenta) become real and can be expressed in terms of one Euclidean form factor as

1 4 kν (2 + iε) = k(2) ln 2C . (2.173) a 0 ≡ 4 l2( 2) − → −

This Wick rotation after moving the second brane to the AdS horizon imposes a special choice of vacuum or special boundary conditions at the AdS horizon. The 2.9 The action for the RS one-brane model 59

Hartle boundary conditions corresponding to this type of analytic continuation imply that the basis function u (z) instead of (2.84) is given by the Hankel function, − (1) 2 2 2 u (z) = H2 (z√ ) = J2(z√ ) + iY2(z√ ) , (2.174) − and thus corresponds to ingoing waves at the horizon [23, 70, 71]. This is equivalent to the replacement Y −, Y − 1, J −, J − i, in (2.169) and, thus, justifies the 1 2 → 1 2 → − Wick rotation of the above type.23 Hartle boundary conditions and the Euclidean form factor (2.173) naturally arise when the Lorentzian AdS spacetime is viewed as the analytic continuation from the Euclidean AdS (EAdS) via Wick rotation in the complex plane of time. Under this continuation the AdS horizon is mapped to the inner regular point of EAdS, the coordinate z playing a sort of an inverse radius, and the regularity of fields near this point is equivalent to the exponential decay of the Euclidean basis function u (z) = − 2iK (z√ 2)/π 0, z (remember that 2 < 0 in Euclidean spacetime). Such 2 − → → ∞ an analytic continuation models the mechanism of cosmological creation via no- boundary [72] or tunneling [99] prescriptions extended to the braneworld context [84, 36, 37, 45]. In particular, it determines (otherwise ambiguous) nonlocal operations in Lorentzian spacetime from their uniquely defined Euclidean counterparts. Wick-rotating the reduced action for large brane separation (2.171), taking the limit a according to (2.172) and (2.173) and rotating back to Lorentzian → ∞ signature we find the non-local braneworld action for the RS one-brane model

2 1 4 l 4 µναβ Sred[gµν , ψ] = d x √g R + Cµναβ ln 2 C + πi C , 16πG4 8 l 2 − Z (2.175) where we have used the on-shell reduction for the radion (2.163). The imaginary contribution to the action obviously makes the effective four-dimensional theory non-unitary. This, however, signifies no physical inconsistency because the imaginary term just corresponds to the loss of energy in form of gravitational radiation into the bulk. This is nothing but a manifestation of the Hartle-Hawking prescription which demands that there are no gravitational waves entering the spacetime from infinity.

2.9 The nonlocal action for the RS one-brane model

Of course, it is possible to obtain the result for the low-energy eﬀective action of the RS one-brane model (2.175) by starting directly with the one-brane setting. The procedure then goes along the same lines as that for the two-brane model. In the one-brane case the boundary conditions at the second brane are to be replaced

23Albeit Hartle boundary conditions can be obtained by a Wick rotation from the large brane separation limit with Neumann boundary conditions on the Σ−-brane, they cannot be obtained by (1) 2 imposing the condition u−(z) = H2 (z√ ) first at a finite position in the bulk (“at the Σ−-brane”), then constructing the two-brane effective action, and finally moving the Σ−-brane — the position at which the boundary conditions are applied — to the horizon. Instead one has to impose Hartle boundary conditions at the horizon right from the start of calculations, so that the second brane is already absent at the level of the equations of motion before constructing the (one-brane) effective action. This indicates that the situation is more involved than described above, but we shall not dwell on these subtleties here. 60 2 The nonlocal braneworld action by some suitable boundary conditions at the horizon of the Poincaré patch of AdS space, y = . Usually these boundary conditions are chosen to be Hartle conditions ∞ due to the arguments given in the previous section. The basis function u (z) of the − Green function therefore takes the form (1) 2 u (z) = cH2 (z√ ) . (2.176) − The homogeneous solution u+(l√2) stays the same as in the two-brane case (2.84) because the boundary conditions at the Σ+-brane remain unaltered. The coefficient c in (2.176) remains undetermined. Restricting both arguments z, z0 to the brane the Green function (2.86) is given in terms of u (l√2), u+(l√2)u (l√2) G(2) = − . (2.177) ∆ The Wronskian inner product of the basis functions is now given by 2 ∆ = c H(1)(l√2) . (2.178) π 1 Here c is the same constant as in (2.176). Therefore the constant c drops out of the Green function and G(2) is given by

1 H(1)(l√2) G(2) = 2 . (2.179) l√2 (1) 2 H1 (l√ ) The corresponding kernel F (2), which is the inverse of the Green function, F (2) = 1 G− (2), is given by H(1)(l√2) F (2) = l√2 1 . (2.180) (1) 2 H2 (l√ ) The limit of low-energies on the brane l22 1 for the Green function and its kernel are obtained by using the relation between Hankel functions of the first kind (1) and Bessel and Neumann functions Hn (x) = Jn(x) + iYn(x) and employing the expansions (2.92)–(2.95), 2 l√2 π G(2) = + ln + C i + O(l22) , (2.181) l22 2 − 2 l22 l422 √2 π F (2) = ln + C i + O(l623) . (2.182) 2 − 4 2 − 2 The nonlocal effective action for the RS one-brane model is obtained by exactly the same procedure as applied to the two-brane model. On the practical side this amounts to dropping all fields related to the Σ -brane in the calculations of Secs. − 2.4 and 2.5 and replacing the F++(2) by F (2) as given by (2.180). The general expression for the nonlocal effective action of the RS one-brane model in terms of the on-brane metric gµν and the radion ψ is 1 2F (2) 1 1 2F (2) S [g , ψ] = d4x R Rµν R + R 4 µν 16πG µν l222 − 3 22 l222 4 Z 1 1 2F (2) 1 3 2ψ + R + 2ψ + R , (2.183) − 6 −2 l222 6 2.9 The action for the RS one-brane model 61 where we have again deliberately singled out a combination of the Ricci scalar and 2 1 the radion ψ + 6 R which vanishes on-shell due to the one-brane analogue of (2.67) 1 2ψ + R = 0 . (2.184) 6 The effective action in the limit of low energies on the brane is found by inserting the expansion for the kernel F (2) (2.182) into the action (2.183)

1 l2 4 S [g , ψ] = d4x R + C ln 2C + πi Cµναβ 4 µν 16πG 8 µναβ l22 − 4 Z 1 l422 4 1 3 2ψ + R ln 2C + πi 2ψ + R , (2.185) − 6 4 l√2 − 6 where we have made use of Eq. (2.158) to transfer the leading order in 2 part of the non-local action into an Einstein-Hilbert action. The leading Einstein-Hilbert term in the effective action reflects the fact that four-dimensional Einstein gravity is recovered in the low-energy regime on the brane (cf. [21]). The result for the effective action is in accord with the one obtained by a Wick- rotation from the limit of infinite brane separation in the two-brane model in the previous section. In particular also in this approach the radion arrises as a dipole ghost which is removed from the action by considering only the on-shell degrees of freedom.

From nonlocal action to other methods

3.1 Eﬀective action of brane-localized ﬁelds vs. Kaluza- Klein reduction

Let us clarify the difference between the Kaluza-Klein and braneworld definitions of the effective actions. In the KK setting the construction of the effective action consists in the well-known procedure of decomposing the multi-dimensional field Φ(x, y), where x are the visible (four-dimensional) coordinates and y are the coordinates of extra dimensions, into a certain complete set of harmonics Zn(y) on the y-space, Φ(x, y) = n φn(x)Zn(y), and substituting the result into the fundamental action S[ Φ(x, y) ] of the field Φ(x, y). Subsequent integration over y, which can be P done explicitly because the harmonics Zn(y) are supposed to be known in explicit form, gives the effective action for an infinite tower of fields φ (x) in x-spacetime, { n } S[ Φ(x, y) ] = dx dy L(Φ(x, y), ∂Φ(x, y)) = dx L ( φ (x) , ∂φ (x) ). (3.1) eff { n } { n } Z Z The harmonics Zn(x) are usually taken as eigenmodes of the y-part of the full wave 2 operator of the theory with eigenvalues mn — the masses of the KK modes. The KK description is particularly justified in the low-energy domain, when only the zero 2 modes n = 0, m0 = 0, are excited, and thus the whole description reduces to a finite number of fields, φ (x) φ (x), — the zero mode KK reduction. This truncation { n } → 0 is efficient when the size of the extra dimensions is small compared to the energy scale of the problem. The zero mode φ0(x) can be regarded as a multi-dimensional field Φ(x, y) averaged in the direction of y and is not associated with the location of the branes in the bulk. Thus, the KK reduction is, in essence, just the change of variables — replacement of one multi-dimensional field by a countable set of four-dimensional fields φ (x) . { n } Dynamically nontrivial input arises only at the level of the low-energy truncation of the above type, when the configuration space reduces to the zero-mode sector φ0(x). As mentioned in Introduction, the action (3.1) is, however, very often not helpful in the braneworld context, because it does not convey a number of its important features like non-compactness of extra dimensions [8, 29], and an interpretation in

63 64 3 From nonlocal action to other methods

terms of the AdS/CFT-correspondence [24, 23, 36]. For example, in the braneworld scenario with non-compact dimensions the spectrum of KK modes becomes continuous, which invalidates the representation (3.1).

3.2 The recovery of the Kaluza-Klein tower

In the next section we reconsider the transverse-traceless sector of the nonlocal braneworld effective action and the equations of motion in terms of a nonlocal kernel and Green function, respectively. In Sec. 3.2.2 to 3.2.4 we derive the localized equations of motion from the nonlocal theory. In Sec. 3.2.2 we present an explicit expansion of the Green function in terms of its residues, which leads to the equations of motion in terms of orthogonal mass eigenmodes of gravitons. In Sec. 3.2.3 we will demonstrate how these modes can be extracted directly from the action. We find that there are two sets of eigenmodes of the action and explain their relation. To elucidate this finding further, we undertake in Sec. 3.2.4 a direct diagonalization of the action in a limit in which the two sets of modes coincide.

3.2.1 The particle interpretation of the transverse-traceless sector In the previous part of this treatise we have derived the eﬀective braneworld action in terms of the four-dimensional on-brane metrics 2 gµν (x) = a ηµν + γµν (x), (3.2) and nonlocal matrix-valued form factors.1 We call the part of the action quadratic in hµν (x) — the transverse-traceless parts of the full metric perturbations γµν (x) on the branes — the graviton sector (cf. Sec. 2.7). It can be written as the following 2 2 quadratic form in terms of the metric perturbations and the special nonlocal × operator F(2), 2 µν 1 4 1 + F( ) h+ Sgrav[ hµν ] = d x [hµν hµν− ] 2 µν . (3.3) 16πG4 2 l h Z − The eﬀective gravitational constant G is given by G G /l. As was shown in 4 4 ≡ 5 Sec. 2.6, the operator F(2) is a complicated non-linear function of the D’Alembert operator 2, expressed by means of Bessel and Neumann functions of arguments l√2 and l√2/a,

2 2 2 1 l√ u+(l√ /a) F(2) = −π , (3.4) + +  2 l√2  −J2 Y2− J2− Y2 √2 − a u (l )  −π −  where  

u (z) = Y1J2(z) J1Y2(z), (3.5) − J + J (l√2), Y + Y (l√2), (3.6) 1 ≡ 1 1 ≡ 1 J − J (l√2/a), Y − Y (l√2/a). (3.7) 1 ≡ 1 1 ≡ 1 1In this chapter we invert our notation for the metric perturbations compared to the previous one: the full metric perturbations on the branes are denoted by γµν whereas the transverse-traceless perturbations are denoted by hµν . 3.2 The recovery of the Kaluza-Klein tower 65

The action (3.3) should be amended by the standard coupling of the transverse- traceless gravitational modes to the transverse-traceless part of the stress-energy tensors, Tµν , on the two branes,

+ 4 + Tµν Smat = d x[hµν , hµν− ] . (3.8) T − Z µν The inverse of the kernel F(2) is the Green function G(2) of the problem,

1 G F− . (3.9) ≡ With the abbreviations given in (3.5) – (3.7) it reads (cf. Sec. 2.6):

l√2 2 a 1 u (l√2) G(2) = a − π . (3.10) 22 + +  2  l J Y − J − Y 2 2 1 1 − 1 1 l√ u+(l√ /a)  π    With the help of G(2) one ﬁnds the equations of motion for the transverse-traceless sector

+ + hµν 2 Tµν = 8πG4l G(2) , (3.11) h− − T − µν µν corresponding to the variation of the combined action of (3.3) and (3.8). In the following we study in detail the properties of the model in the low-energy limit when l√2 1 but when l√2/a can take arbitrary values in view of the smallness d/l of the parameter a = e− (large interbrane distance). As we will only be considering the transverse-traceless part of the gravitational dynamics, tensor indices will be omitted in Secs. 3.2 and 3.3.

3.2.2 The eigenmode expansion of the Green function Both the Green function G(2) and the kernel of the action F(2) are highly nonlocal. Nevertheless, it is possible to obtain a conventional interpretation of the equations of motion and the action in terms of an infinite tower of orthogonal Kaluza-Klein modes. We will first discuss the recovery of the KK tower from the Green function because this procedure works in a more direct way than the recovery of the particle spectrum from the action, which will be discussed in the Sec. 3.2.3. The elements G ( = ) of G(2) are meromorphic functions of 2. If we consider a concentric ×× × circle Cn around 2 = 0 so that Cn includes the first n poles, and let its radius R as n , we find the falloff property n → ∞ → ∞

G (2) < Rn , Rn , (3.12) | ×× | → ∞ for any small constant and all 2 on Cn. Therefore we can employ the Mittag-Leﬄer expansion, which provides an expansion of a meromorphic function in terms of its poles, for the Green function to obtain a representation of G(2) as a sum of scalar propagators. 66 3 From nonlocal action to other methods

In the limit l√2 1 but l√2/a arbitrary the Green function G(2) is given by2 2 2 2 1 π Y1[l√ /a] l√ a 1 l22 2 2 + C + ln l√2 − − J1[l√2/a] 2 J1[l√2/a] G(2)   . (3.13) ≈ 1 J [l√2/a]  a a 2   l√2 2 l√2 2   J1[l√ /a] − J1[l√ /a]    For convenience we use 2 as the fundamental variable of the expansion and not √2. The Mittag-Leffler expansion for an element G of the Green-function matrix ×× reads 2 2 2 ∞ Res[G ( = mi )] G (2) = ×× , (3.14) 2 2 2 ×× mi [ mi ] Xi=0 − where the first pole is at 2 = 0, i. e. m0 = 0. One can write 2 2 2 2 ∞ Res[G ( = mi )] ∞ Res[G ( = mi )] G (2) = ×× + ×× . (3.15) 2 2 2 ×× mi mi Xi=0 − Xi=0 As the second sum contributes only a constant, we can drop this part and obtain our final result for the elements of the Green function, 2 2 ∞ Res[G ( = mi )] G (2) = ×× . (3.16) 2 2 ×× mi Xi=0 − In this way each nonlocal element of the nonlocal Green function G(2) can be represented as an infinite sum of scalar propagators with different masses. The mass- squares are given by the position of the poles of G(2). From the representation of the Green function (3.13) we immediately infer that all elements of G(2) have poles at the same values of 2. This is a direct consequence of the common prefactor displayed in (3.10), which contains all poles of G(2). One pole is located at 2 = 0, which is also an exact pole of (3.10). An infinite sequence of poles is located at the roots of the Bessel function J1. Denoting the argument at the ith zero of J1 by ji we find that these poles are located at 2 = (j a/l)2 m2, (3.17) i ≡ i where the first few values of ji are given by j 3.832 , j 7.016 , j 10.173 . (3.18) 1 ≈ 2 ≈ 3 ≈ Whereas the first term of the sum (3.16) from the pole at 2 = 0 will provide us with the effective four-dimensional massless graviton, the infinite series of poles at mi is responsible for the generation of the KK tower of massive gravitons. The matrix of residues of G at the pole 2 = 0 is given by 2 1 a2 Res[G(2 = 0)] = , (3.19) l2(1 a2) a2 a4 − 2 Note that the falloff property (3.12) is not valid for the element G++ in the approximation (3.13) but only for the full expression (3.10). This does of course not hinder us to use the approximation (3.13) for an approximate determination of the residues and therefore for an approximate determination of the Mittag-Leffler expansion. 3.2 The recovery of the Kaluza-Klein tower 67

2 2 and the residues for the inﬁnite sequence of poles at = mi are found to be 2a2 1/(J [lm /a])2 1/J [lm /a] Res[G(2 = m2)] = 2 i − 2 i . (3.20) i l2 1/J [lm /a] 1 − 2 i It is interesting to note that the residues can be factorized as

T Res[G(2 = mi)] = vivi , (3.21) with

√2 1 √2a 1/J2[lmi/a] v0 = ; vi = , i 1. (3.22) l √1 a2 a2 l 1 ≥ − − This property reﬂects the fact that the residues of the poles of the Green function are projectors on the corresponding propagating modes of the theory. These energy eigenmodes will be recovered from the action in the next section. Before that we should, however, write down explicitly our newly acquired representation of the Green function:

2 1 a2 G(2) = l22(1 a2) a2 a4 − ∞ 2a2 1/(J [lm /a])2 1/(J [lm /a]) + 2 i 2 i . (3.23) 2 2 2 − l ( mi ) 1/(J2[lmi/a]) 1 Xi=1 − − 3.2.3 The spectrum and the eigenmodes of the eﬀective action It is instructive to understand how the eigenmodes of the equations of motion can be recovered from the action (3.3) and its kernel F(2) (3.4). We will ﬁnd that the situation has many similarities with resonance theory, and our discussion will parallel many of the considerations of [100] from which we will also adapt our nomenclature of eigenstates. In the low-energy limit, l√2 1 but l√2/a arbitrary, the kernel of the action F(2) can be approximated as

1 1/J2[l√2/a] 22 2 l F( )  2 J [l√2/a]  . (3.24) ≈ 2 1/J [l√2/a] 1 2 2 2  −l√ a J2[l√ /a]    The typical way to extract the particle content from an action with a matrix- valued kernel is its diagonalization in terms of normal modes. However, as found by the application of the Mittag-Leﬄer expansion of the Green function in the last section, the number of propagating modes enormously exceeds the number of entries in the 2 2-matrix F(2) and the modes therefore do not diagonalize the quadratic × action (3.3) in the usual sense. This behavior could also have been anticipated from the nonlocality of F(2). The propagating modes are the zero modes of F(2) which solve the matrix-valued nonlocal equation

F(2)hi(x) = 0 . (3.25) 68 3 From nonlocal action to other methods

These modes are conveniently split into a scalar part hi(x) depending on the four- dimensional spacetime coordinates and a spacetime-independent (isotopic) vector part vi, hi(x) = hi(x)vi . (3.26) The eigenvector equation (3.25) is accompanied by its consistency condition,

det F(2) = 0, (3.27) which corresponds to picking the poles of the Green function G(2). The condition (3.27) yields the mass spectrum of the theory given by the roots of this equation, i. e. 2 2 = mi , so that the hi(x) above are Klein-Gordon modes and the isotopic vectors 2 of the propagating modes vi are zero eigenvectors of F(mi ),

(2 m2) h (x) = 0, (3.28) − i i 2 F(mi )vi = 0. (3.29)

Thus we obtain the Kaluza-Klein spectrum which contains the massless mode i = 0, m0 = 0, and the tower of massive modes. In the low-energy approximation of F(2), Eq. (3.24), their masses mi = aji/l are given by the roots of the first-order Bessel function, J1(ji) = 0. The isotopic structure of their vi is given by the vectors (3.22) which were found to factorize the residues of the Green-function.3 In view of standard arguments of gauge invariance, the massless graviton h0 has two dynamical degrees of freedom, while the massive tensor field hM posesses all five polarizations of a generic transverse-traceless tensor field. The action (3.3) is not, however, diagonalizable in the basis of these states because under the decomposition h(x) = i=0 hi(x)vi (with off-shell coefficients hi(x)) the cross terms intertwining different i-s are non-vanishing, P vT F(2)v = 0. (3.30) i j 6 A crucial observation is, however, that the diagonal and non-diagonal terms of this expansion are linear and bilinear, respectively, in on-shell operators 2 m2, − i 2 T T dF( ) 2 v F(2)vi = v vi (2 m ), (3.31) i i 2 2 2 i d =mi − vT F(2)v = M (2)(2 m2)(2 m2), i = j, (3.32) i j ij − i − j 6 2 2 2 where higher powers of ( mi ) have been dropped in (3.31), and Mij( ) is non- 2 −2 2 2 vanishing at both = mi and = mj . Therefore, the non-diagonal terms of the action do not contribute to the residues of the Green function G(2) of F(2). The normalization of vi as given by (3.22) automatically yields a unit coefficient of (2 m2) in (3.31). − i Thus we have found the particle spectrum of the action in terms of modes which turn out to be non-orthogonal off-shell but become orthogonal if one considers the

3 2 T The vector v0 is here actually given by v0 = √2/l[1 a ] , i. e. the leading order in a of the exact v0 as given by (3.22), because here the vi’s are obtained for the approximation (3.24) of F. 3.2 The recovery of the Kaluza-Klein tower 69 action on the mass-shell. In the theory of atomic and nuclear resonances such non- orthogonal energy eigenstates are called Siegert states after their introduction in [101] (cf. [100]). This particle spectrum can also be recovered from the two orthogonal eigenstates of F(2). These states which will be energy dependent, i. e. not mass eigenstates, have also been extensively used in resonance theory and are named Kapur-Peierls states according to their ﬁrst use in [102, 103] (cf. also [100]). In resonance theory, Kapur-Peierls states bi(2) diagonalize the matrix Hamilto- nian of the system (the analogue of our F(2)). For our 2 2 matrix-valued kernel × F(2) this property reads

F(2)bs(2) = λs(2)bs(2) , s = 1, 2, (3.33)

T 2 0 2 0 bs ( )bs ( ) = δss , (3.34) where the second equation describes the orthonormality of the Kapur-Peierls states. In the limit l√2 1 the energy-dependent eigenvalues of F(2) [as given by Eq. (3.4)] are found to be

2 l 2 l√2 J1[lmi/a] λ1(2) = + O(a) , λ2(2) = + O(a) . (3.35) 2 a J2[lmi/a] The corresponding eigenvectors (normalized to unity to leading order in a) are

1 1 1 b1(2) = a l√2 , b2(2) = , (3.36)   √1 Z2 Z 2J1[lmi/a] − −   where we have introduced l√2 2 1 Z + J [lm /a] J [lm /a] . (3.37) ≡ a l22 2 1 i − 2 i

2 The eigenvalue λ1 given in (3.35), has only one root at the value = 0, whereas the eigenvalue λ2 has infinitely many zeros at the zeros of the Bessel function J1, i. e. 2 2 2 at = mi . Approximating λ2( ) around its zeros by the first term of its Taylor expansion we find for the eigenvalues l22 l2 λ (2) = , λ (2) = (2 m2) . (3.38) 1 2 2 2a2 − i

At 2 = 0 the eigenvector b1 takes the value 1 b (0) = , (3.39) 1 a2 2 2 and at energies = mi the eigenvector b2 becomes 1 b (m2) = . (3.40) 2 i J [lm /a] − 2 i As expected, the mass levels of the Siegert states equal the zeros of the Kapur- Peierls eigenvalues. The massless graviton Siegert state corresponds to the single 70 3 From nonlocal action to other methods

root of the eigenvalue λ1(2), and the tower of massive Siegert states arises from the inﬁnitely many zeros of the eigenvalue λ2(2). It is easy to ﬁnd the relation between the Siegert eigenvectors and the Kapur-Peierls eigenvectors by noting from 1 Eq. (3.33) that the vectors bs(2) are eigenvectors also of F− (2) = G(2) with eigenvalues 1/λs(2). Using the completeness and orthonormality of the bs(2)’s we obtain a representation of G(2),

b (2)bT (2) G(2) = s s . (3.41) λs(2) sX=1,2

As the poles of the Green function lie at λs(2) = 0 we ﬁnd for the residues of G(2) 2 T 2 2 2 2 bs( )bs ( ) lim ( mi )G( ) = . (3.42) 2 m2 − dλs(2)/d2 2 2 → i =mi

where s = 1 for m0 0 and s = 2 for mi with i 1. Comparing this expression for ≡ ≥ the residues with (3.21) we obtain

1/2 1/2 dλ (2) − dλ (2) − v = 1 b (2) , v = 1 b (2) , i 1. 0 d2 1 i d2 2 ≥ 2 2 2 =0 =mi

(3.43) The decomposition of the action into Siegert eigenmodes (3.26 ) provides us with the conventional particle interpretation of the propagating modes of the nonlocal operator F(2) and clarifies their role in its Green function (3.23) mediating the gravitational effect of matter sources. Amended by the matter action on the branes the effective action of the graviton sector reads 2 4 1 T F( ) 1 T S [ h ] = d x h h + h T , (3.44) µν 32πG l2 2 Z 4 where T is the column vector of the transverse-traceless part the stress-energy tensors on the branes, and h is now given by the sum of Siegert modes (3.26),

h = hi(x)vi . (3.45) Xi=0 Varying this action with respect to each Siegert mode hi and recomposing the results to recover the equations of motion for h, we obtain the linearized equations of motion, their solution h = 8πG l2G T (3.46) − 4 ret being expressed in terms of the retarded version of the Green function (3.23). We + note the explicit expressions for h and h− obtainable from (3.43) and (3.45):

+ √2 1 1 h (x) = h0(x) + a hi(x) , (3.47) l √1 a2 J2[lmi/a] ! − Xi1 √2 a2 h−(x) = h0(x) a hi(x) . (3.48) l √1 a2 − ! − Xi1 3.2 The recovery of the Kaluza-Klein tower 71

It is particularly interesting to study the asymptotic behavior of the expressions (3.47) and (3.48). In the long-distance limit a 0, we have →

h+ h + O(a) , (3.49) ∼ 0 2 h− a h0 a hi . (3.50) i 1 ∼ − ≥ X + Thus, in this limit the h -mode practically coincides with the massless graviton h0, and the mixing with the other modes is suppressed by the factor a. On the other hand, the Σ -brane mode h− is almost exclusively composed from the massive modes − and mixing with the massless mode is suppressed, whereas the mixing between the massive modes can be called maximal, i. e. h− is proportional to the sum of the massive modes. Moreover, the contribution of both the modes hi and h0 to the h−- mode in this regime is suppressed by the common factor a. There is also a regime of maximal mixing between all modes for the metric perturbation h− on the Σ -brane − at a = 1/√2 where

h− h0 + hi . (3.51) i 1 ∼ ≥ X

3.2.4 The graviton eﬀective action in the diagonalization approximation

In the last section we have found that the Siegert state representing the massless graviton corresponds to the zero of the eigenvalue λ1 of the kernel of the action F(2), whereas the Siegert states corresponding to all of the KK modes stem from the zeros of the second eigenvalue λ2. This opens the possibility of a low-energy limit in which the Siegert and Kapur-Peierls states coincide. In the approximation

√2 1 , l/a√2 . 4 (3.52) one can use the small-argument expansion for all Bessel and Neumann functions appearing in F(2). If we truncate this expansion at O(2) we obtain an expression for the kernel F(2) which is entirely local. The choice of the second part of the low-energy condition (3.52) guarantees that the approximation is still valid slightly above the mass of the ﬁrst KK mode,

m = j a/l lm /a 3.83 , (3.53) 1 1 ⇔ 1 ≈

(cf. 3.18). In fact, at least for Bessel functions of the second kind the small-argument expansion is considered as the method of choice in numerical calculations for arguments . 4 [104]. However, we should be prepared for some deviation of the expansion-result for the ﬁrst KK mode from that of the previous sections due to the low order of truncation of the expansion. Taking this into account the result will turn out surprisingly accurate. In the low-energy approximation (3.52) the kernel F/l2 can be represented in 72 3 From nonlocal action to other methods

the form [56]:

F(2) = M + D2 + O(22), (3.54) l2 − 1 8 a4 a2 M = − , (3.55) l2 1 a4 a2 1 − − 1 a2 a2 + 3 2 D = − . (3.56) 3(1 + a2)2 2 3 + a 2 − The operator D represents the kinetic part of the action (3.3), while the operator M plays the role of the mass term. Calculating the trace and the determinant of the matrix D, it is easy to see that it is positive definite for any value of the parameter a. The determinant of the matrix M is equal to zero, therefore M is degenerate. Using the positive-definiteness of the matrix D, one can diagonalize both the matrices D and M simultaneously. Our purpose here is not simply to diagonalize these matrices, but also to make a canonical normalization of the kinetic terms. With this choice the action for gravity eigenmodes with fixed masses, representing some mixtures of the functions h+ and h−, will have the form

1 1 1 S [h , h ] = d4x h 2h + h (2 M 2)h , (3.57) graviton 0 M 16πG 4 0 0 4 M − M 4 Z where M is the mass of the massive gravitational mode. The details of the procedure of diagonalization are presented in the Appendix. Here we shall give the ﬁnal results. The expression for the mass of the massive graviton mode hM is 24a2(1 + a2) M 2 = . (3.58) l2(1 a2)2 − It is easy to see that at small values of the parameter a, i. e. at large distances between the branes, the squared mass behaves as

24a2 M 2 (3.59) ∼ l2 and tends to zero as a2. This result is to be compared with the more exact expression for the ﬁrst KK mass from the relation (3.17). The large discrepancy is easily explained from the fact that taking the limit a 0 is strictly speaking not legitimate → in our approximation because the second condition of (3.52) would not be fulﬁlled anymore. The relations describing the transition from the old graviton variables h+ and h− to the modes hM and h0 have the following form,

√1 a2 h h = − ah+ − , (3.60) M √ 2 − a 3(1 + a ) 2 √1 a + h = − (h + h−). (3.61) 0 1 + a2 3.3 Digression: radion-induced graviton oscillations 73

The inverse transformations are

+ 1 h = (h0 + √3ahM ), (3.62) √1 a2 a− h− = (ah0 √3hM ). (3.63) √1 a2 − − The corresponding expressions for v0 and vM are √2 1 √6a 1 v0 = , vM = , (3.64) l√1 a2 a2 l√1 a2 1 − − − which should be compared with the expressions for v0 and v1 from (3.22). As might have been anticipated from the fact that the residues depend only on the ( 1)th − term of the Laurent expansion of the function considered, we ﬁnd exact agreement for v0. Amending the action (3.57) by the matter action (3.8) we can obtain the equa- + tions of motion for the metric perturbations h and h− on the brane by varying the total action with respect to h0 and hM and then combining their equations of motion according to the transformations (3.62) and (3.63),

h+ 1 1 1 a2 = 16πG4 2 2 4 h− − 2 1 a a a ret − 2 + 1 3a 1 1 T + − . (3.65) 2 M 2 1 a2 1 1 T ret − − − −

In the language of the Siegert and Kapur-P eierls eigenmodes of Sec. 3.2.3 we have chosen for the diagonalization an approximation in which the Siegert eigenmodes become orthogonal and in which the Kapur-Peierls eigenvalues and eigenstates become energy-independent. Thus in the approximation (3.52) the Siegert and Kapur-Peierls descriptions coincide. The ﬁrst Kapur-Peierls state coincides with the Siegert state of the massless graviton, while the massive Siegert states become a kind of “collective” massive state which coincides with the second Kapur-Peierls mode.

3.3 Phenomenological digression: radion-induced graviton oscillations

3.3.1 Overview An effect so far overlooked in previous studies of braneworld models is the mixing between graviton modes of different masses, which inevitably occurs in higher- dimensional spacetime models. This effect is of little relevance in models with small extra dimensions and heavy KK modes, because in these models the massive gravitons will rarely be produced and therefore are of no astrophysical significance. In models where gravitons can have masses in the sub-eV scale like in the RS models, the mixing between the different modes can lead to interesting effects in gravitational-wave astrophysics. Gravitational waves (GW’s) will then consist of contributions from several graviton modes. This will induce a beat in the gravitational wave which could be detectable with gravitational-wave interferometers. The 74 3 From nonlocal action to other methods

parameters of the oscillations will crucially depend on the size of the extra dimension — commonly parameterized by the radion-field —, thus suggesting the notion of radion-induced graviton oscillations (RIGO’s). RIGO’s are particularly interesting because their observability is not limited to a certain minimal value of the extra-dimensional curvature scale. RIGO’s could therefore become the first detectable signature of M-theory phenomenology. From the very beginning the affinity of the RS two-brane setup to the M-theory compactification of [105] (see [106] for reviews of effective five-dimensional M-theory) was noticed [7]. Actually, the RS two-brane model can be viewed as a simplified version of this M-theory solution, which appears five-dimensional in an intermediate energy-range. The RS model simplifies the effective supergravity action of [105] to an Einstein- Hilbert action and neglects the gauge-field part of the M-theory effective action. Due to this relationship, the RS model has been used as an approximation for aspects of M-theory cosmology where a full-fledged M-theory calculation is not feasible (see [40] for an example). On the other hand the RS model deviates in major aspects from the requirements of M-theory: first, in contrast to the RS model, M-theory restricts the size of the largest extra dimension to typically 10 31 m l 10 29 m [107, 106] (see, however, − ≤ ≤ − [108]). This rules out the possibility to detect the extra dimension by measuring short-distance deviations from Newtonian gravity. Second, in M-theory the requirements of having the standard-model group structure and supersymmetry breaking, forces one to identify our Universe with the positive-tension brane (cf. [42]). This prevents one from obtaining the simple geometrical solution to hierarchy problem by locating our Universe on the negative tension brane of the RS model.4 Therefore it seems that one has to choose between either the attractive phenomenology of the RS model or the appealing theoretical framework of M-theory, although it remains tempting to conjecture that if M-theory realizes large extra-dimensions in nature they should exhibit a similar setup as the Randall-Sundrum two-brane model. Considering RIGO’s one may, however, be able to probe several additional orders of smallness of the extra-dimensional curvature radius if only the radion acquires suitable values. Although the required values of the radion are still far from the parameter range of typical M-theory models, the search for RIGO’s therefore offers a possibility to narrow the gap between the phenomenologically accessible parameter space of large extra dimensions and the predictions of M-theory. In the preceding chapter we have studied the four-dimensional effective action in the RS two-brane model. This action is a functional of the two induced metrics and radion fields on the two four-dimensional branes. Using the results of [60, 59] we shall here extend the study of radion-induced graviton oscillations in the two-brane world of [60] and investigate their phenomenological consequences. The section will be devoted entirely to phenomenology: we discuss the description of RIGO’s and their possible observational consequences in the RS model. After reminding the reader of the analogous effect of quantum oscillations in Sec. 3.3.3 we first discuss the phenomenology of RIGO’s on the positive-tension brane in Sec.

4There may remain some loopholes though, see for example [109] for a construction, which would allow us to live on the positive-tension brane and have the hierarchy inherited from the negative- tension brane, or [110, 111] for M-theory constructions which would allow the standard-model ﬁelds to reside on the negative-tension brane. 3.3 Digression: radion-induced graviton oscillations 75

3.3.4. This will lead us to consider in particular the gravitational waves produced by a network of cosmic strings on the negative-tension brane in Sec. 3.3.5. In Sec. 3.3.6 follows the discussion of RIGO’s on the negative-tension brane. Finally, in Sec. 3.3.7 the discussion is extended to so-called bi-gravity braneworld models. In the concluding Sec. 3.3.8, we summarize our results and argue for the occurrence of RIGO’s in a general class of braneworlds featuring orbifolded compactiﬁcations. A detailed presentation of the diagonalization procedure for the kinetic and massive parts of the gravitational eﬀective action used in Sec. 3.2.4 is given in the Appendix.

3.3.2 Gravitational waves on the Σ+-brane After the extensive theoretical considerations of Secs. 3.2.3 and 3.2.4 we will now return to the equations of motion (3.11) and study their phenomenological properties. We will proceed to show how graviton oscillations arise naturally in the propaga- tions of gravitational waves in this model. For studying the properties of graviton oscillations it is convenient to abandon the condensed matrix notation of Eq. (3.11) and study the behavior of metric perturbations on the Σ+-brane and Σ -brane − separately, starting with the Σ+-brane. Using the spectral representation (3.23) we ﬁnd for h+,

2 2 + 16πG4 + 2 ∞ 16πG4 a + a h = (T + a T −) T T − , (3.66) − 2 − 2 m2 2 − ret i=1 i ret 2 2 X − J J where we have intro duced J [lm /a] 0 .403. We consider astrophysical J2 ≡ 2 1 ≈ sources of equal intensity at x = 0 on both branes with a harmonic time dependence iωt T (t, x) = µe− δ(x) , (3.67) where µ is the quadrupole moment of the source. The sources with an oscillating quadrupole moment will generate gravitational waves. At a distance r from the source the waves on each brane are given by a mixture of massless and massive spherical waves. On the Σ+-brane this superposition is given by the sum of the contributions

2 a 2 2 + + iωt iωr i√ω mi r h [ T ] = A e− e + 2 e − , (3.68) (J2[lmi/a]) ! Xi=1 + 2 iωt iωr 1 i√ω2 m2r h [ T −] = Aa e− e e − i , (3.69) − J2[lmi/a] ! Xi=1 from the sources on the Σ+- and Σ -brane, respectively, where A = 4 G4 µ/r is the − amplitude of the massless mode produced on the Σ+-brane. If the frequency ω with which the source is oscillating is smaller than the mass mi of a KK mode, ω < mi, then the exponent of the corresponding mode will be real and negative instead of imaginary, and the mode will be decaying instead of oscillating. In effect all modes with masses above the frequency of the source will have died off after propagating for a short distance and only those modes with a mass below the frequency of the source will contribute to the gravitational wave in the far field. Thus as long as we 76 3 From nonlocal action to other methods

are interested in GW’s we can content ourselves with including only the terms with mi < ω in the sums of Eqs. (3.68) and (3.69) (cf. [112]). In order to have a simple situation we consider a source with a frequency above the mass threshold of the ﬁrst massive mode but below the threshold of the second KK mode, i. e. m1 < ω < m2. Then only the massless and the ﬁrst massive mode are excited and produce long-range gravitational waves. At a distance r from the source the waves on each brane are given by a mixture of massless and massive spherical waves. On the Σ+-brane this superposition is given by the sum of the contributions

2 a 2 2 + + iωt ikr i√k m1r h [ T ] = A e− e + 2 e − , (3.70) J2 + 2 iωt ikr 1 i√k2 m2r h [ T −] = Aa e− e e − 1 , (3.71) − J2 from sources on the Σ+- and Σ -brane, respectively. −

3.3.3 Quantum oscillations — an analogy The above results have interesting physical consequences. We have seen that the observable transverse-traceless metric on each brane is indeed a linear combination of different gravitational modes with fixed masses. One of these modes appears to be massless, while the others are massive with their masses depending on the distance between the branes. This situation is quite analogous to the well-known mixing arising in the context of kaons [113, 114], of other bosons [115], or neutrino oscillations [116]. As a matter of fact, the phenomenon which we encounter here is not quantum oscillations but classical ones. Nevertheless, the interference effects between the different gravitational modes, arising in the process of their propagation in space (on the brane), are analogous to those of quantum oscillations. This is quite natural because the oscillations of quantum particles are totally conditioned by their wave nature (see, for example, Ref. [114]), and both effects are caused by the superposition of modes with different dispersion relations. However, there is one important difference: it does not matter for quantum oscillations if one considers the superposition to consist of two components of different energies but with the same momentum (as is usually assumed), or if the components are supposed to have the same energy but different momenta [117]. For a mixture of classical gravitational waves, however, it is natural to assume that they have been produced by the same source oscillating with a frequency ω. Therefore one should consider oscillations between modes of the same frequency ω, which have different wave-numbers k, and not vice versa. Let us recall briefly some basic formulas from the theory of quantum oscillations. Suppose that a certain particle can exist in two different observable states ψ and | 1i ψ , which correspond to a quantity conserved in some interaction but which do not | 2i have fixed masses. For example, kaons can be produced or detected in states with fixed strangeness, which are not eigenstates of their Hamiltonian, and according to the theory of neutrino oscillations, which is now supported by good experimental evidence [118], the states of the electron, muon and tau neutrinos are also mixtures 3.3 Digression: radion-induced graviton oscillations 77 of different Hamiltonian (i. e. mass) eigenstates. Generally, this situation can be described for the two-state case by

ψ = cos θ ψ + sin θ ψ , | 1i | M i | mi ψ = sin θ ψ + cos θ ψ , (3.72) | 2i − | M i | mi where the states ψ and ψ are states with masses M and m, respectively, and | M i | mi the angle θ parameterizes the mixing between the states. If at the initial position of an evolution at x = 0 somebody observes, say, a particle in a state

ψ(0) = ψ , (3.73) | i | 1i propagating in space with the energy ω, then at the position x the state (3.73) will become equal to

ikM x ikmx ψ(x) = cos θ e− ψ + sin θ e− ψ | i | M i | mi i√k2+M 2x i√k2+m2x = cos θ e− 0 ψ + sin θ e− 0 ψ , (3.74) | M i | mi where k0 = ω. Now we are in a position to calculate the probability of observing the particle in the state ψ at the position x, which is given by the formula | 1i (k k )x P (x) = ψ ψ(x) 2 = 1 sin2(2θ) sin2 M − m . (3.75) 1 |h 1| i| − 2 Correspondingly, the probability of ﬁnding the particle in the state ψ is | 2i (k k )x P (x) = ψ ψ(x) 2 = sin2(2θ) sin2 M − m . (3.76) 2 |h 2| i| 2

It is easy to see that the oscillations of the probabilities P1 and P2 are strongest when sin2(2θ) = 1, i. e. when π θ = . (3.77) 4 This situation is called the case of maximal mixing between oscillating particles. Looking at the expressions for the probabilities (3.75) and (3.76), one can see that they represent periodic functions with a period 2π L = . (3.78) k2 + M 2 k2 + m2 0 − 0 This is the oscillation length of pa quantum statep with the energy ω = k0. In the case of maximal mixing (3.77), detecting the particle ψ at the distance | 1i L/2, one ﬁnds with probability P (L/2) = 1 the particle in the state ψ . In other 2 | 2i words, the particle in the state ψ is totally transformed into the particle in the | 1i state ψ . The time-dependence of the oscillation can be inferred from this ex- | 2i pression by semi-classically considering the propagation speed of the corresponding particles. The description of quantum oscillations presented above coincides with describing classical oscillations of superpositions of waves with diﬀerent dispersion relations. In this case, instead of the probability of detecting a particle of a certain type, one can calculate the amplitude of the wave. 78 3 From nonlocal action to other methods

3.3.4 Gravitational-wave oscillations on the Σ+-brane For the superpositions (3.68) and (3.69) the amplitudes detected by a gravitational- wave interferometer are given by the absolute values of the waves,

4a2 2 πr 1/2 h+[ T +] = + 1 J2 sin2 , (3.79) A − ( 2 + a2)2 L J2 1/2 + 4 2 2 πr h [ T −] = − 1 + J sin . (3.80) A ( 1)2 L 2 J − Here, L is the oscillation length of the amplitude modulation of the gravitational wave (GW), 2π L = . (3.81) ω ω2 m2 − − 1 This corresponds to Eq. (3.78) with one mop de being massless. The pre-factors of the amplitudes (3.79) and (3.80) are given by

a2 + = 1 + A A, (3.82) A 2 ≈ J2 1 2 2 − = 1 a A 1.5 a A, (3.83) A − ≈ J2 where the approximations are valid in the limit a 1. We ﬁnd oscillations in the amplitudes of the waves from both sources. For a GW produced by T +, Eq. (3.79), the oscillating part of the amplitude is suppressed by a factor of a2 compared to the constant part in the limit of large brane separation, a 1. The amplitude of the GW produced by T −, Eq. (3.80), is totally oscillating, regardless of the inter-brane distance. In the limit of a source-frequency just above the mass-threshold ω & m1 the oscillation length (3.81) of these radion-induced gravitational (“graviton”) wave oscillations (RIGO’s) tends to 2π L & . (3.84) m1 This limit is mainly useful to set a lower bound on the oscillation lengths as in this limit the propagation speed of the massive mode will tend to zero. Therefore in this limit the massless and massive wave trains from a source would soon become spatially separated from each other and far from the source one would detect two separated wave trains instead of one wave train exhibiting RIGO’s. The astrophysically more interesting limit is the case ω m in which the expression for the oscillation length 1 becomes ω L = 4π 2 . (3.85) m1 We can express the minimal oscillation length (3.84) of RIGO’s through the AdS radius l and the scale factor a,

l l Lmin = 2π 1.6 , (3.86) j1a ≈ a 3.3 Digression: radion-induced graviton oscillations 79

where j1 is the first root of J1 (cf. (3.17) and (3.18)). The oscillation length is inverse proportional to a. Graviton oscillations become observable when the oscillation length is at least of the order of the arm length of a GW detector. For the ground-based interferometric detectors this requirement corresponds to L 103 m. ∼ Combining this with the constraint on the maximal size of the AdS radius l from 4 sub-millimeter tests of gravity, l . 10− m [119], we find an upper limit on the warp 7 factor a . 10− for the oscillation length to be detectable. Inserting this into the ratio of the amplitudes (3.82) and (3.83) we find

+ 2 14 −/ a . 10− . (3.87) A A ∼ Therefore, the amplitude of a wave originating from a source on the (“hidden”) Σ -brane with oscillations which are sufficiently long to be detectable, is strongly − suppressed by a damping factor a2 compared to a GW stemming from a source on the Σ+-brane itself. A strongly oscillating wave has to be generated by a source 14 orders of magnitude stronger than that of a weakly oscillating one in order to be of the same magnitude, which at first sight makes the detection of RIGO’s impossible. We will, however, see in the next subsection that the damping factor could easily be overcome if the RS model is considered as a model for a putative M-theory realization of large extra dimensions. Beforehand we want to summarize that the main differences between the graviton oscillations considered here and the more traditional neutrino or kaon oscillations: First, as is readily inferred from Eq. (3.45) or (3.47) and (3.48), respectively, the + transformation between the gravitational modes h and h− living on certain branes and the graviton modes h0 and hi possessing fixed masses is not an orthogonal (and not a unitary) transformation. This is connected with the fact that — in contrast to the mixing for neutrinos or kaons, where only the mass matrix is non-diagonal if written in terms of observable states — in the case of gravity in a two-brane world the whole kernel of the action is non-diagonal and, moreover, is diagonalizable only on-shell (cf. Sec. 4). The second and, perhaps, more important distinguishing feature of RIGO’s is the dependence of the mixing parameters and the mass of the massive mode on the parameter a, i. e. their dependence on the radion. If one considers a model in which the distance between the branes is time-dependent, the mixing parameters and all the oscillatory effects become functions of time depending on the particular features of the model under consideration.

3.3.5 High-amplitude RIGO’s on the Σ+-brane from M-theory The bad prospects for the detection of RIGO’s found at the end of the last section may considerably be improved if one considers the RS two-brane model as a toy model for a yet-to-be-constructed model of the strong-coupling limit of heterotic M-theory with one large extra dimension. Although such a model is lacking, we can extrapolate generic features of M-theory compactiﬁcations with small extra dimensions and draw some reasonable conclusions for the hidden sector of the model. These conclusions suggest the possibility that the hidden brane could in fact be the source of GW’s with very large amplitude. 80 3 From nonlocal action to other methods

Cosmological M-theory models consist of an 11-dimensional spacetime which is 1 compactified on an S /Z2 orbifold with a stack of D-branes on each orbifold-fixed plane [120, 121, 105].5 Further 6 of the 10 dimensions are compactified on a Calabi- Yau three-fold, while the other dimensions parallel to the stacks of branes, including the timelike direction, remain uncompactified. The compactification scale of the Calabi-Yau manifold is generally assumed to be smaller by one or two orders of magnitude than the size of the compact 11th dimension. Each stack of branes hosts one super-multiplet of E8 gauge fields. This construction can become a suitable model for our universe if one of the set of E8 fields is broken into the subgroup E6 and further into SU(3) SU(2) U(1). These fields are then called the visible sector × × of the model and the corresponding stack of branes is identified as our universe, while the other E8 is called the hidden sector. In order to obtain a viable phenomenology it is necessary to break the E8 of the visible sector by gaugino condensation in the hidden sector and then mediate the influence of the gaugino condensate to the visible sector by its coupling to moduli fields on the Calabi-Yau manifold.6 The volume of the Calabi-Yau space V at the visible sector is assumed to be of the size of the standard-model grand-unification scale V 1/6 1016 GeV. In generic − ∼ M-theory compactifications it has been found necessary in this context to choose the volume of the Calabi-Yau space at the hidden sector smaller than at the visible sector, which will also lead to a stronger gauge coupling in the hidden sector than in the visible one [121] (cf. also [42]). Also the volume of four-dimensional slices parallel to the uncompactified dimensions of the stacks of branes will decrease towards the hidden sector. It has been observed that in order to obtain a viable phenomenology one has to assume that the E8 on the hidden sector is also broken to a smaller group [122]. In order to make connection of these M-theory setups to the RS two-brane model one identifies the visible stack of branes with the Σ+-brane and the hidden stack of branes with the Σ -brane: the orbifolded coordinate of M-theory is identified with − orbifolded bulk coordinate of the RS model, and the Calabi-Yau space is neglected in the effective five-dimensional description due to its small volume [7]. The major deviation of the RS model from M-theory ideas is that the size of the orbifolded bulk may be large. In M-theory models the scale factor for the uncompactified coordinates corresponding to the warp factor a(y) in the RS model is tied to the Calabi-Yau volume. A decreasing warp factor also leads to a decrease of the Calabi-Yau volume. In the M-theory solution of [105] the relation between the four-dimensional scale factor and the Calabi-Yau volume reads, for example,

V (y) a6(y) , (3.88) ∼ where y denotes the coordinate of the orbifolded dimension. In scenarios with a large orbifolded dimension the dependence of the Calabi-Yau volume on the scale factor should be weaker because the string scale sets a lower limit for the size of the Calabi-Yau manifold. Nevertheless, if M-theory can be extrapolated to setups with a large bulk one has to expect a considerable decrease of the Calabi-Yau volume

5cf. [42] for a non-technical description of the cosmological aspects of this class of models. 6 cf. [1], Chap. 18.3, p. 366–371, for a pedagogical description of this mechanism in the weak- coupling limit of E8 E8 heterotic string theory. × 3.3 Digression: radion-induced graviton oscillations 81 from the visible to the hidden brane. The crucial observation for our argument is now that the vacuum expectation value of the gaugino condensate η located on the hidden brane depends strongly on the inverse size of the Calabi-Yau space [123],

α3 V η exp 18π (S βT ) . (3.89) ∼ V 9/2 − b α − 0

Here, α is the 10-dimensional gauge-coupling constant, b0 is the beta-coefficient of the gauge-coupling renormalization group, S and T are moduli of the Calabi-Yau space and β describes loop-corrections to the moduli fields. From the structure of (3.89) we find that even a moderate decrease of V with the warp factor leads to a strong increase of the vacuum expectation value of the gaugino condensate, which will therefore be very large in a RS-like model deduced from M-theory. Turning now to the theory of cosmic strings, it is well known that the quadrupole moment of a cosmic string is proportional to the square of the vacuum expectation value η of the corresponding symmetry breaking,

µ Γη2/ω3 , (3.90) ∼ where Γ 50 . . . 100 is a numerical coefficient depending on the trajectory and ≈ shape of the string loop and ω is the characteristic frequency of string oscillations [124]. Therefore the large gaugino vacuum expectation value η would lead to a very large quadrupole moment for cosmic strings produced during the breaking of the hidden E8, which in turn will produce high-amplitude gravitational waves on the hidden brane. The described effect may easily compensate the damping factor (3.87) and lead to strong gravitational waves from the hidden brane also on the Σ+-brane. These in turn will then distinguish themselves from waves produced on the Σ+-brane itself by detectable RIGO’s. Note that the described mechanism is not spoiled by the dependence of the effective gravitational coupling constant on the volume of the Calabi-Yau space, G G /V , because the dependence of the gaugino vacuum expectation value on 5 ∼ 11 the size of the Calabi-Yau space is much stronger than that of the effective five- dimensional gravitational coupling. At first glance our result about a strongly enhanced symmetry-breaking vacuum expectation value on the hidden brane compared to the visible brane seems to be in sharp contrast to the conventional interpretation of the solution to the hierarchy problem in the RS-model (cf. e. g. [7, 9, 109]). This conventional interpretation states that the vacuum expectation value of a symmetry breaking on the hidden brane should be suppressed by two powers of the warp factor compared to the same value on the visible brane, 2 η = a η+ . (3.91) − However, already Randall and Sundrum pointed out that the generation of the hierarchy could equally well be described by considering the gravitational scale on the Σ -brane as the derived scale, whereas the symmetry breaking scale on both − branes is to be taken the same [7]. That this interpretation is in fact the correct one has first been demonstrated in [94] by considering the fall-off properties of on-brane Green functions in the Euclidean domain and has been confirmed by an analysis 82 3 From nonlocal action to other methods

of the measuring process for masses on the Σ -brane in [69]. Therefore, all eﬀects of the hierarchy generation in the RS model are− already contained in the damping factor (3.87) and there is no geometrical suppression of symmetry-breaking scales on the hidden brane in the RS model.

3.3.6 Graviton oscillations on the Σ -brane − Graviton oscillations on the Σ -brane can be treated along the same lines as RIGO’s − on the Σ+-brane have been treated in Secs. 3.3.2 and 3.3.4. Abandoning once again the matrix notation we ﬁnd for h− with the help of the spectral representation (3.23) the equations of motion

2 16πG4 2 + 4 16πG4 a + 2 h− = (a T + a T −) T + a T − . (3.92) − 2 − 2 m2 − ret i=1 i ret 2 X − J

Considering the same set of sources (3.67) we now study the gravitational waves seen by an observer on the negative tension brane,

+ 2 iωt iωr 1 i√ω2 m2r h−[ T ] = Aa e− e e − i , (3.93) − J2[lmi/a] ! Xi=1 2 iωt 2 iωr i√ω2 m2r h−[ T −] = Aa e− a e + e − i . (3.94) ! Xi=1 We again suppose for simplicity that only the massless and the ﬁrst massive modes are produced, i. e. m1 < ω < m2. Then we have for the waves produced by the source on the Σ+-brane and the Σ -brane, respectively, −

+ 2 iωt iωr 1 i√ω2 m2r h−[ T ] = Aa e− e e − 1 , (3.95) − J2 2 iωt 2 iωr i√ω2 m2r h−[ T −] = Aa e− a e + e − 1 , (3.96) where A = 4 G4 µ/r is the amplitude of the massless mode produced on the Σ+- brane. The amplitudes detected by a gravitational-wave interferometer are given by the absolute values of (3.95) and (3.96),

1/2 + + 4 2 2 πr h−[ T ] = 1 + J 2 sin , (3.97) A ( 2 1) L J − 2 1/2 4a 2 πr h−[ T −] = − 1 sin . (3.98) A − (1 + a2)2 L Here, L is the oscillation length of the amplitude modulation of the gravitational wave, given by (3.81). The pre-factors of the amplitudes (3.97) and (3.98) are

1 + = 1 a2A 1.5 a2A , (3.99) A − ≈ 2 J 2 2 2 − = (1 + a )a A a A . (3.100) A ≈ 3.3 Digression: radion-induced graviton oscillations 83

Therefore we have for the ratio of the typical intensities measured on the Σ -brane, − + A 1.5 . (3.101) ≈ A− For waves on the Σ -brane we thus find that the intensities of waves produced on this − brane and on the other brane are of the same magnitude. Therefore at first glance the situation of an observer on the Σ -brane seems to be favorable for observing graviton oscillations. − Unfortunately, this effect is spoiled by the dependence of the graviton mass — and therefore of the oscillation length — on the position of the brane. In order to find the oscillation length actually measured by an observer on the brane we have to use the coordinate system in which an observer does his measurements. The coordinates of an observer will always be chosen with respect to the Lorentz metric ηµν (cf. [69]). No rescaling is necessary for studying effects on the Σ+-brane because there the induced metric coincides with the Minkowski metric. On the Σ -brane the − “physical” coordinates used by an observer are related to those given with respect 2 to the induced metric gµν = a ηµν by

µ µ xphys = ax . (3.102)

In the equations of motion (3.11) the ﬂat space d’Alembertian has to be replaced by the one with respect to the physical coordinates,

2 µν ∂ ∂ 2 µν ∂ ∂ 22 = η µ ν = a η µ ν = a phys. (3.103) ∂x ∂x ∂xphys ∂xphys

To be precise we would also have to rescale the metric perturbation hµν with respect phys 2 to the background metric gµν to the measured one hµν = hµν /a (y) with respect to the Lorentz metric. We will not conduct this step because we are comparing amplitudes measured on the same brane and therefore this rescaling does not change the ratio between the amplitudes. In physical coordinates the source-free equations of motion on the Σ -brane are given by − 2 phys m phys 2 h0 = 0, 2 i hi = 0, (3.104) − phys µν − phys − a2 µν whereas the equations of motion on the Σ+-brane remain unchanged. Therefore the graviton mass appears diﬀerent when observed on a diﬀerent brane,

+ a mi ji m = m = j , m− = = . (3.105) i i i l i a l

This rescaling is characteristic of bulk fields in the RS two-brane model [69, 94].7 + Whereas the measured graviton masses mi on the Σ+-brane depends on the warp factor, the mass mi− on the Σ -brane is independent of the inter-brane distance. − 7Note that the rescaling does not apply to on-brane fields and therefore does not affect the symmetry breaking scale considered in Sec. 3.3.5 (cf. [69, 94]). 84 3 From nonlocal action to other methods

From these masses we can derive the oscillation length observed on the branes for a

GW with a wave-number ω & mi−. In this limit we have 2π L = . (3.106) m1− On the Σ -brane this becomes − 2π l L− = = 2π . (3.107) m1− j1

4 By this expression the upper experimental limit on the AdS radius l . 10− m also yields the upper limit on the oscillation length on the Σ -brane. However, an − oscillation length of sub-millimeter size is clearly unobservable. Therefore graviton oscillations will be hidden to an observer on the Σ -brane. − To conclude, the situation is unfavorable for the detection of RIGO’s by an observer on the Σ -brane: albeit being connected with strong GW’s the observer − cannot detect the oscillations because their oscillation length is far too short to exhibit itself in interferometric detectors.

3.3.7 RIGO’s in bi-gravity models Bi-gravity models are braneworld setups in which the effective four-dimensional gravity on the brane which is considered as our Universe is mediated by one massless and one light massive gravitational mode which have approximately equal strong couplings to matter on the visible brane. The rest of the KK spectrum is separated by a mass gap and its coupling to matter on the visible brane is strongly suppressed. Meanwhile there are various realizations of these setups [15, 16], although only the six-dimensional ones do not seem to suffer from a VvDZ discontinuity for the massive mode or from a phenomenologically unrealistic negative curvature of the visible brane. For our consideration we need not consider a specific realization of these models but can content ourselves with their general structure of the equations of motion and the assumption that phenomenological constraints from the VvDZ discontinuity are somehow circumvented. Then the four-dimensional effective equations of motion for the transverse-traceless sector of a generic bi-gravity model reads

c c c hvis = 16πG 0 + 1 + n T vis , (3.108) 2 2 2 2 2 − " ret ret m1 ret mn # − nX=2 − e where

c c , c c , n 2, (3.109) 0 ≈ 1 n 0 ≥ m1 ∆mn , ∆mn = mn mn 1 , n 2 (3.110) − − ≥ and we have only included sources T vis on the visible brane. The eﬀective gravitational constant G does not coincide with the usual eﬀective four-dimensional gravitational constant. Rather we have e G G(c + c ) . (3.111) 4 ≈ 0 1 e 3.3 Digression: radion-induced graviton oscillations 85

From (3.109) we infer that we have a strong mixing between the massless mode and the first KK mode. This would make bi-gravity models a natural candidate for strong RIGO’s. However, this expectation is spoiled by phenomenological constraints on this class of models. The major experimental constraint is established by precision measurements of Newton’s law from the orbital motion of the planets. Our treatment will closely follow that of [125] where a purely massive gravitational interaction is considered. It is sufficient to consider only the influence of the first KK mode, since the heavier modes will only contribute to the short-range dynamics. The non- relativistic static gravitational potential V (r) = G M/r then gets modified by a − 4 Yukawa potential to the form

GM r V (r) = c0 + c1 exp , (3.112) − r −λ1 e where M is the mass of the central body. The gravitational acceleration of a test body in the modiﬁed potential is g = e µ(r)/r2 where e is the unit-vector pointing − r r from the central body to the test body and

µ(r) = GM [c + c (1 + r/λ ) exp ( r/λ )] 0 1 1 − 1 c r 2 r 3 = GMe c + c 1 + O . (3.113) 0 1 − 2 λ λ " 1 1 # e For pure Newtonian gravity µ = G4M = constant and its value can be determined to a high precision from the orbit of the earth around the sun. If a massive gravitational mode contributes, the µ determined for the orbits of other planets will diﬀer from the earth-orbit value. For a planet with a semi-major axis ap and a period Tp, Kepler’s 1/3 2/3 third law yields µ (ap) = ap (2π/Tp). Therefore it is convenient to parameterize the deviation of a planet’s µ(ap) from the one of the earth µ(a♁) through a small parameter ηp, 1/3 µ(ap) 1 + ηp . (3.114) ≡ µ(a♁)! Using (3.113) in (3.114) we obtain a lower bound on the wavelength of the ﬁrst KK mode in terms of the experimental limits on ηp,

2 2 1/2 c1[a♁ ap] λ1 = − . (3.115) 6ηp(c0 + c1)!

Restricting to the case that c c 1/2 we ﬁnd 0 ≈ 1 ≈ 2 2 1/2 a♁ ap λ1 = − . (3.116) 12ηp !

The most restrictive bound on ηp comes from the measurement of the Mars orbit for which we have η < 6.5 10 10 [126]. This leads to the requirement m − × − m 2π/λ < π/(1012 km) (3.117) 1 ≡ 1 86 3 From nonlocal action to other methods

for a realistic bi-gravity model. One might try to infer more stringent upper bounds on the mass of the first KK mode by considering the motions of galaxies and galaxy- clusters. However, these considerations rely heavily on the assumed amount of dark matter in the universe. In view of the fact that the cosmology of braneworld models is still in its infancy and can provide us with quite surprising models of dark matter (see e. g. [127]) we desist from using such constraints, which would lead us to an upper mass for the first KK mode close to the bounds on the graviton mass given 19 1 by the Particle Data Group, mg . (10 km)− [128]. However, even the less restrictive solar-system limit (3.117) on m1 renders RIGO’s in bi-gravity models unobservable. Inserting (3.117) into the formula for the oscil- 4 lation length (3.85) we find even for GW’s of a frequency of 10− Hz, the lowest frequency observable by LISA, an oscillation length of more than 300 light years. An amplitude modulation of that length will clearly remain undetectable. The limit on the wavelength of the first KK mode does however still allow for the possibility of detecting ultra-light KK modes by studying the propagation speed of gravitational waves with LISA (cf. [125]). In addition to the bounds on the length of RIGO’s from solar-system dynamics it is interesting to note that some models even predict a minimal oscillation length beyond the present-day Hubble radius of our Universe. This is in particular the case in the Kogan-Mouslopoulos-Papazoglou version [16] of the Karch-Randall model [13] with two AdS4 branes embedded in AdS5. This model is particularly attractive because the longitudinal polarization of the KK gravitons is suppressed in the propagator by the geometry of the setting and therefore the VvDZ discontinuity is absent [16, 14]. In this model the strongest constraint on the length of RIGO’s comes from the requirement that the observable cosmological constant of our Universe, i. e. the effective cosmological constant, Λ4, of one of the AdS4 branes should be compatible with the observed magnitude of the cosmological constant,

120 4 Λ . 10− M , (3.118) | 4| Pl where MPl denotes the four-dimensional Planck mass. This requirement on the geometry of the model is reflected in the coefficients of the harmonics of the bulk spacetime and thereby restricts the mass of the first KK-mode m1 to a fraction of 1 the inverse size of the Hubble radius of our Universe, H − , 135 1 m e− /(H− ) . (3.119) 1 ∼ Using the expression for the lower limit of the oscillation length (3.84), the value for the first KK mass leads to an oscillation length for RIGO’s,

135 1 1 L e H− H− , (3.120) min ∼ which is much larger than the size of the horizon of our Universe and thus renders RIGO’s unobservable in principle in this particular model.

3.3.8 Summary on RIGO’s From the equations of motion for the transverse-traceless modes of the RS model we have found a mixing of massless and massive modes which depends parametrically 3.4 Holographic interpretation 87 on the radion. This mixing leads to the effect of radion induced graviton oscillations, which is in principle observable. RIGO’s are a feature of every higher-dimensional spacetime model, since there will always occur amplitude modulations in GW’s, which are a mixture of a massless mode and KK modes. However, in traditional models with flat extra dimensions, the mass of the first KK mode is so big that it will neither be produced by astrophysical sources nor lead to oscillation lengths of macroscopic size. In contrast to this, warped geometries allow KK mode masses which are so low that they can lead to oscillations of detectable length. In particular, waves from sources on the hidden brane show strong oscillations on the visible brane. The amplitudes of these waves are strongly suppressed by the warped geometry and therefore at first sight seem to remain undetectable. However, by using simple M- theory motivated scaling arguments we have demonstrated that one should expect a network of cosmic strings on the hidden brane which would produce a background of high amplitude gravitational waves. The M-theory scaling properties considered may easily compensate the geometrical damping of GW’s from the hidden brane. Therefore GW’s with RIGO’s stemming from the hidden brane may actually become a relevant effect for gravitational-wave astronomy. The characteristic pattern of RIGO’s may even help in the discrimination between noise and the stochastic GW background for GW interferometers. At first sight another natural candidate for observable RIGO’s are bi-gravity models. In these models the massless graviton and the lightest massive one are coupled to matter with nearly equal strength and, therefore, produce strong oscillations. Unfortunately, bi-gravity models which exhibit RIGO’s of experimentally detectable oscillation-length are already ruled out by precision measurements of Newton’s law on solar-system scales. It is worth mentioning that the effects connected with the mixing and oscillations of quantum states in a multidimensional spacetime have received some attention in the literature [129, 130, 131, 132]. Neutrino mixing and oscillations were reconsidered in [129, 130, 131], while the mixing between quarks and an attempt to explain the origin of CP violation was described in [132]. Nevertheless, to our knowledge, the possibility of oscillations between different graviton states and their potential observability in GW interferometers has not been considered before.

3.4 Holographic interpretation of the nonlocal action

3.4.1 The RS one-brane model In Chap. 2 we have stressed at several occasions that our results can be interpreted as a manifestation of the AdS/CFT correspondence. In particular the form factors found in the limit of large brane separation show a logarithmic term as the leading nonlocality which is characteristic for the AdS/CFT correspondence. Indeed, the connection of the nonlocal eﬀective action to the AdS/CFT correspondence principle extends much further. The nonlocal action of the RS one-brane model is in fact identical to the action of the model derived from the AdS/CFT correspondence in [24, 133] (cf. Sec. 1.3.2.1). This is easily demonstrated by transforming our eﬀective action obtained in Sec. 2.9 to the conventions employed in Sec. 1.3.2.1. Leaving the position of the Σ -brane, z = l exp( y /l), arbitrary and restricting the radion to + + | +| 88 3 From nonlocal action to other methods

its on-shell value, the eﬀective action for the one-brane model reads

z2 z2 4 S [g , ψ] = + d4x R + C ln 2C + πi Cµναβ . (3.121) 4 µν 16πG l 8 µναβ z22 − 5 Z In this expression we followed the practice of Chap. 2 by raising indices not with the 2 2 induced metric on the brane gµν = (l /z+) ηµν but with the Minkowski metric ηµν . In order to compare the action with the AdS/CFT results discussed in Sec. 1.3.2.1 we transform the curvature and the D’Alembert operator to the form, that they take 8 if one raises indices with the induced metric, gµν ,

2 4 l µν l µν R = 2 , Rµν = µν , R = 4 , (3.122) z+ R R z+ R 2 2 2 l µν l 2 = 2 g µ ν = 2 . (3.123) −z+ ∇ ∇ z+ The action then takes the form e l 1 4 S = d4x√g + ln 2C + πi µναβ . (3.124) 16πG R 8Cµναβ l22 − C 5 Z This expression is identical to the action (1.39), obtained in [24] by the AdS/CFT e calculations described in Sec. 1.3.2.1, apart from a factor of 2 arising due to the Z2- orbifold symmetry that was assumed throughout Chap. 2 but not in the AdS/CFT calculation of Sec. 1.3.2.1 (cf. the remark on p. 22, Sec. 1.3.2.1). In contrast to the AdS/CFT calculation, which does not determine the renormalization scale in the form factor of the Weyl-tensor-squared term, this form factor is now completely fixed. Thus, our method independently recovers the AdS/CFT description of the RS one-brane model in terms of the stress-energy tensor of two putative CFT’s, Eq. (1.40), coupled to four-dimensional gravity. Furthermore, it extends the applicability of the AdS/CFT description to calculations beyond the recovery of effective four-dimensional Einstein gravity by fixing the kernel of the second-order-in- curvature term. This term is particularly interesting because it will be responsible for the deviations from Einstein gravity in the cosmological evolution of the RS model.

3.4.2 The RS two-brane model For the RS two-brane model no action has yet been constructed by means of holographic methods. In the present section we develop a generalization of the procedure [74, 24] discussed in Sec. 1.3.2.1. The eﬀective action on the Σ+-brane was obtained by substituting a solution of the bulk Einstein equations into the bulk action and subsequent integration over the extra dimension. The eﬀective action on the brane arose as a boundary term of the integration. In a straightforward generalization one can introduce the Σ -brane as a second boundary of the AdS space without mod- − ifying the procedure itself. The second brane then produces a second surface term

8We do not need to consider the transformation of the Riemann tensor because in Eq. (3.121) we can single out the Gauss-Bonnet invariant, cf. Eq. (2.161), without changing the variational derivative of the action. 3.4 Holographic interpretation 89

which contributes to the effective gravitational action on the Σ+-brane. In this way on should recover the reduced effective action for the fields on the Σ+-brane. Since this procedure does not distinguish between the branes the action will be valid also for the Σ -brane. The gravitational parts of the surface terms for the two branes − have to be conformally equivalent up to the calculated order. The couplings to matter and therefore the effective gravitational constants will, however, be different on the two branes. By introducing the Σ -brane as a second boundary, one expects to recover the − correct form of the transverse-traceless sector of the action only. All contributions of the radion will be absent because one treats the branes as stiff boundaries which do not allow for brane-bending and thus one freezes the radions. However, it is possible to reintroduce the radion by considering linearized perturbations of the interbrane distance. One might anticipate a more serious problem though: the calculation of the boundary term at the Σ+-brane relied on the specific expansion of the metric about the coordinate singularity at ρ = 0, Eqs. (1.34) and (1.35). This expansion is valid only for a limited range of values of the ρ-coordinate and therefore might break down before the position of the Σ -brane is reached. This would render the proposed − method for the calculation of the holographic two-brane action impracticable. The range of validity of the expansion (1.32) has been tentatively addressed in [133]. There it was suggested that it should be possible to integrate along the ρ coordinate until the bulk volume element vanishes. Thus the integration should be valid until some value ρcrit, for which

2 µν det A(x, ρcrit) = 0 , where A (x, ρ) = g(0)gµν , (3.125)

with g(0)µν , gµν as introduced in Eq. (1.35). As A(x, ρ), in general, has more than one root it remains unclear up to which of the zeros the integration will remain valid (cf. [133]). For our purposes it is sufficient that the integration will be possible up to some finite distance from the Σ+-brane. In this way we will be able to recover the reduced action on the Σ+-brane for small interbrane distance. Our starting point is, just as in Sec. 1.3.2.1, the five-dimensional Einstein-Hilbert action [cf. Eq. (1.29)]. This time, it is accompanied by two boundary terms — one for each brane —,

1 S [G ] = S +S = dd+1x√G (R 2Λ) grav MN bulk boundary 16πG − d+1 Z 1 1 + ddx g K + α 8πG ind(+) (+) 2 (+) d+1 ZΣ+ p 1 d 1 + d x gind( ) K( ) + α( ) . (3.126) 8πG − − 2 − d+1 ZΣ− p

Into this action we substitute the bulk metric (1.34) and integrate over the ρ coordinate. Denoting the position of the Σ -brane as ρ = /a2, a being the RS warp factor − − 90 3 From nonlocal action to other methods

deﬁned as usual (cf. Sec. 1.2), we ﬁnd the two-brane analog of the action (1.36),

2 1 4 /a √g 1 8 4 S = d4x dρ + + ρ∂ + α √g 16πG l ρ3 ρ2 − l l ρ (+) 5 ZM4 Z ρ= 1 8 4 + ρ∂ρ + α( ) √g , (3.127) ρ2 l − l − ρ=/a ! where the integration over the bulk coordinate now runs from to /a. Analogously to the one-brane case, Eq. (1.30), α(+) and α( ) are given by −

16πG5 α(+) = σ(+) + Lmat (gind(+), φ+) , (3.128) √gind(+) 16πG5 α( ) = σ( ) + Lmat (gind( ), φ ) , (3.129) − − √gind( ) − − − where gind( )µν are the induced metrics and Lmat( ) are the Lagrangian densities of matter ﬁelds φ on the respective branes. We also use the convention for the brane tension introduced in Sec. 1.3.2.1 in which the RS values of the tensions become σ = 6/l. Expressing gµν in the action (3.127) by the expansion (1.35) and performing the integration as in Sec. 1.3.2.1 we obtain

1 1 6 lα S = d4x g α + gµν g 16πG (0) 2 (+) − l 2 (0) (2)µν 5 ZM4 " 3 p 4 4 5 2 l µναβ l a 6 l a α µν + ln C(0)µναβ C(0) g(0) 2 α( ) + + g(0)g(2)µν 16 # − " − l 2 3 p l µναβ + ln 2 C(0)µναβ C(0) + terms, ﬁnite for 0 . (3.130) 16 a # → ! In Eq. (3.130) we have already dropped the Gauß-Bonnet invariant from the curvature-squared term because it will not contribute to the variation of the action (cf. p. 56). If we express the action in terms of the induced metric on the brane at 2 , γµν = (l /)gµν , the boundary contribution of the Σ -brane will be the same as − that of the Σ+-brane, but rescaled by a conformal transformation with the conformal factor a2. We ﬁnd

2 l 4 l µναβ S = d x √γ + ln µναβ 32πG5 M4 R 8 C C Z 2 2 l µναβ 4 √γ a + ln µναβ + . . . + Smat (+) + a Smat ( ) , (3.131) − R 8 a C C ! − where the actions of the matter ﬁelds on the branes,

4 4 Smat (+) = d x Lmat(γ, φ+) , Smat ( ) = d x Lmat(γ, φ ) (3.132) − − Z Z 3.4 Holographic interpretation 91

have been extracted from α(+) and α( ) respectively (note that also the Lagrangian density for matter on the Σ -brane has− been formulated with respect to the induced − metric γµν on the Σ+-brane) and the RS values of the brane-tensions, σ = 6/l, have been inserted. The dots in Eq. (3.131) indicate terms of higher orders in the derivatives of the induced metric (cf. Sec. 1.3.2.1). We can combine the contributions of the two boundaries to yield

l l2 1 S = d4x√γ (1 a2) + ln µναβ + . . . 32πG − R 8 a2 CµναβC 5 ZM4 4 + Smat (+) + a Smat ( ) . (3.133) − In this expressions the two logarithms have merged into a quantity which is dimensionless without introducing any renormalization scale. Eq. (3.133) coincides with the leading order in a of Eq. (2.162), found by reducing the braneworld eﬀective action in Sec. 2.8, in the transverse-traceless sector. It lacks, however, the nonlocal radion-term, (a2/6)R(1/2)R, because we did not take into account the bending − of the branes. The brane-bending mode can be introduced in a heuristic way as a small variation of the warp factor. Above we have found that the boundary terms on the branes are conformally related. Thus, it is not necessary to introduce two radion modes and we can restrict our considerations to the bending of the Σ+-brane. The radion can be introduced into the warp factor as follows:

2d/l 2[d+ξ(x)]/l 2 a = e− e− a (1 2ξ(x)) . (3.134) → ≈ − Using the redeﬁnition (2.76), ψ(x) ξ/l, and the equations of motion of the radion, ≡ Eq. (2.67), 62ψ + = 0 , (3.135) R we can reconstruct the corresponding kinetic term of the radion Lagrangian. It reads

L = 3ψ2ψ + ψ . (3.136) radion R Accounting for the radion, the action for stiﬀ branes (3.133) gets modiﬁed to

l S = d4x√γ (1 a2) + 2a2(3ψ2ψ + ψ ) 32πG − R R 5 ZM4 2 h l 1 µναβ 4 + ln µναβ + . . . + Smat (+) + a Smat ( ) . (3.137) 8 a2 C C − i We can remove the explicit dependence of the holographic action on the radion by tracing out the radion at tree level. This leads to

l a2 1 S = d4x√γ (1 a2) two-brane 32πG − R − 6 R2R 5 ZM4 2 h l 1 µναβ 4 + ln µναβ + . . . + Smat (+) + a Smat ( ) . (3.138) 4 a2 C C − i In Eq. (3.138) we ﬁnd the same nonlocal expression for the brane-bending degree of freedom as in Eq. (2.162) of Sec. 2.8.1. The action (3.138) derived by holographic 92 3 From nonlocal action to other methods

considerations just recovers the leading behavior in the warp factor a of the reduced action for small brane separation (2.162) obtained from the two-field action by tracing out the fields on the hidden brane. Beyond being a consistency check, the comparison of the nonlocal braneworld action with the holographic result leads the way to applications of the nonlocal method to the AdS/CFT correspondence. We present one particular aspect by checking the claim of [80] where the cutoff induced by the Σ -brane was interpreted as a sponta- − neous symmetry breaking in the CFT dual of the RS model (cf. Sec. 1.3.2.2). The effect of the negative-tension brane is easiest investigated if one first removes the ultraviolet anomaly of the CFT by moving the Σ+-brane to ρ = 0 so that only the infrared symmetry breaking of the CFT persists. If the type of symmetry breaking induced by the Σ -brane were indeed spontaneous, a conformal variation of the brane position should− not contribute to the stress-energy tensor of the CFT (cf. [80]). This can be checked by varying the contribution of the action arising from the boundary at ρ with respect to conformal rescalings of the metric. If the variation − is non-vanishing then the second boundary contributes to the conformal anomaly of the dual CFT. In order to study the behavior of the Σ -brane part of the action − under conformal variations δϕ we note that g(0)µν and behave as

δg(0)µν = g(0)µν δϕ , δ = δϕ . (3.139)

Using these transformation properties we can perform the conformal variation of the Σ -part of the action, −

2 2 l 4 a 6 a δ d x g(0) α + R − 16πG l2 − − l2 2 ( 5 ZΣ− " p 2 l µναβ + ln 2 C(0)µναβ C(0) + . . . 16 a #) l l2 = d4x g C Cµναβ + . . . δφ , (3.140) 16πG (0) 16a2 (0)µναβ (0) 5 ZΣ− p where the dots indicate higher order terms in . Although the square of the Weyl tensor is a conformal invariant the total variation is non-vanishing due to the variation of ln(/a2). Thus, the surface term from the second brane modifies the conformal anomaly of the dual CFT. This modification of the Weyl anomaly clearly contra- dicts the assumption that the insertion of the Σ -brane only leads to a spontaneous − breaking of the CFT. Thus in contrast to the claims [80], the IR cut-off in the CFT by the Σ -brane induces a soft symmetry breaking in the CFT. − 4

Conclusions and Outlook

In the present work we have constructed the braneworld effective action for the two- and one-brane models of the Randall-Sundrum type up to quadratic order in curvature perturbations and radion fields on both branes. We have obtained the exact nonlocal form factors and their low-energy approximation. The form factors for the two-brane model have an infinite series of zeros which are identifiable with the masses of the Kaluza-Klein modes. Based on this observation we established a method to extract the particle content from the nonlocal braneworld effective action. Thus, the nonlocal description is intrinsically equivalent to the KK setup. However, it explicitly features two metric fields rather than the infinite tower of local KK modes. The price one pays for this is that the two-field action is essentially nonlocal, the nonlocality being a cumulative effect of the KK modes. For the RS two-brane model we have also considered the reduced version of the action — the functional of the fields associated with only one positive-tension brane. A physical motivation for this reduction is the fact that, if this brane with its metric and other fields is regarded as the only visible one, then one has to trace out the fields on the second brane in the whole two-brane system1. In the tree-level approximation this procedure is equivalent to the exclusion of the invisible fields via their equations of motion in terms of the fields on the positive-tension brane. In the low-energy approximation the result turns out to be very simple — the action is dominated by the Einstein term with the four-dimensional gravitational constant, which explicitly depends on the brane separation. This is a manifestation of the well-known localization of the graviton zero mode on the brane, i. e. the recovery of the low-energy Einstein theory. However, for finite interbrane distance the action does not get completely localized even in the long-distance approximation — in the conformal sector a certain nonlocality survives. Part of this nonlocality can be localized in terms of the additional scalar field non-minimally interacting with the brane curvature. The result coincides with the braneworld action constructed in [45] by a simplified method disregarding the metric perturbations on the negative-tension brane. In contrast to [45], the radion field in (2.162) has a dipole ghost nature, but on-shell the action (2.162) coincides with that of [45], given by (2.165). Moreover, off-shell both actions can be identically transformed into one another by the nonlocal reparameterization,

1This may also lead to decoherence for the visible ﬁelds, cf. [140].

93 94 4 Conclusions and Outlook

(2.164). Thus, as a by-product our results justify the conclusions of [45]. There these results were used to generate inflation by means of the radion field (2.164) playing the role of an inflaton. For this purpose, the model was generalized to the case when the tension on the visible brane is slightly detuned from the flat-brane RS value. Then the action (2.165), rewritten in the Einstein frame, would acquire a nontrivial inflaton potential, such that its slow-roll dynamics corresponds to branes diverging under the action of a repulsive interbrane force — a scenario qualitatively different from the models of colliding branes [38, 40, 82]. However, it suffers from an essential drawback: the necessity to introduce by hand a four-dimensional cosmological constant — the brane tension detuning of the above type. The present results suggest a different mechanism of interbrane repulsion based on the presence of the Weyl-squared term in (2.162) and (2.171). When the brane 2 Universe is filled with gravitational radiation, this term is nonzero, Cµναβ > 0. For small brane separation it forms the interbrane potential (l2/2)κ (a)C2 . − 1 µναβ The potential has a maximum at the point of coinciding branes a = 1 because the coefficient κ1(a) given by (2.154) is strictly positive. The repelling force is very small, though, and identically vanishes at a = 1, because of the behavior of κ1(a) at the brane collision point2, κ (a) (1 a2)3/12. Unfortunately, this potential is strictly 1 ∼ − negative, because κ (a) 0, and, therefore, cannot maintain inflation (for recent 1 ≥ studies of models with a negative cosmological constant, see [141]). Rather, it can serve as a basis of brane models with AdS4 geometry embedded in AdS5 [13]. It can also be useful in the model of “thick” branes in the Big Crunch/Big Bang transitions [142] of ekpyrotic and cyclic cosmologies [40, 83]. Thus, the renormalization flow in the brane distance a (AdS flow) can, in principle, be realized in our model at the dynamical level. This flow, interpolating between short and long interbrane distances, has remarkable properties. It features a transition from the phase of the local action (2.165) to the action (2.171) with logarithmic nonlocal form factors in the Weyl-squared term. At the initial stage the local logarithm in κ1(a), Eq. (2.154), structurally resembles the logarithmic behavior of the Coleman-Weinberg potential [143]. The scalar field in the potential is that defined in Eq. (2.164), ϕ 3/4πG a, and the potential reads ∼ 4 κ (pa) = (1/4) ln(ϕ2/m2 ) + . . . , m2 = 1/G . (4.1) 1 − P P 4 However, in contrast to the original Coleman-Weinberg Lagrangian the potential multiplies the square of the Weyl tensor instead of ϕ4. When extrapolated to small a (big interbrane distances) it enters, instead of a naive infinite growth, another energy domain (2.137), a2 l22, where it gets saturated by the characteristic scale ∼ of the Weyl tensor (or the energy scale of the gravitational radiation contained in the model). Therefore, the local coefficient κ1(a) gets replaced by the form factor k1(2), Eq. (2.169). Hence the dominant logarithmic term, κ (a) (1/4) ln(1/a2) , (4.2) 1 ∼ goes over into the logarithm k (2) (1/4) ln(4/l22) . (4.3) 1 ∼ 2Interestingly, the expression (2.154) represents the logarithm ln(1/a2) with exactly the first two terms of its Taylor series in (1 a2) subtracted. − 4 Conclusions and Outlook 95

Thus, this renormalization AdS flow leads to the delocalization of the initial radion 2 2 µναβ condensate κ1(a)Cµναβ to nonlocal (short-distance) corrections Cµναβ k1( )C , characteristic of the AdS/CFT-correspondence principle in the limit of large interbrane distance. Concrete implications of these phase transitions in cosmology still have to be worked out. Here we only remark that they can comprise an essential point of depar- ture from the scenario of diverging branes [45] for large interbrane distances (2.137). The corresponding action (2.171) differs from (2.165) [or equivalently (2.162)], which was extrapolated in [45] to late stages of the brane runaway. In contrast to (2.165), the action (2.171) does not have any non-minimal curvature coupling of the modulus a, which together with the brane tension detuning served as a basis for the acceleration stage in [45]. The diverging-branes scenario [45] also admitted this stage as a sequel to the slow-roll inflation, but the inflation and acceleration stages overlapped there and, thus, caused unsurmountable difficulties for reheating [141]. It would be interesting to observe the effect of the nonlocal short-distance corrections (replacing the non-minimal curvature coupling of (2.165)) on the late time behavior in the scenario of diverging branes. Since these corrections are dominated by curvature- squared terms, their effect can be equivalent to the R2-inflation model [144] (see also [139] for the same conjecture on the realization of the Starobinsky model in braneworld scenarios). The nonlocal action we found for the one-brane model coincides with that found by considerations based on the AdS/CFT correspondence [24] [our Eq. 1.39]. How- ever, we were able to improve the AdS/CFT result of [24] because in our method it was possible to fix the renormalization scale which was still left open in the AdS/CFT calculations. Our result for intermediate brane separation casts a new light on the CFT-dual interpretation of the second brane in the RS model. Previously it has been claimed that the insertion of the second brane leads to spontaneous breaking of dilatation invariance of the AdS bulk [80, 145]. This would correspond to a spontaneous breaking of the conformal invariance of the CFT. The breaking takes place in the infrared because the second brane is inserted into the high-energy region of the bulk and should therefore, according to the UV/IR correspondence, induce an infrared effect in the CFT. As a proof for the spontaneous nature of the symmetry breaking it is stated in [80] that the stress-energy tensor of the CFT remains unmodified by the insertion of the brane. The modifications of the CFT only arise due to the CFT-equivalent of the radion which is the Goldstone mode arising from the breaking of dilatation invariance. In contrast to these claims we find a difference between the kernel of the Weyl-tensor-squared part of the one-brane model and the corresponding term in the two-brane model for finite brane distance. Since the Weyl-tensor squared term generates the stress-energy tensor of the CFT (cf. Sec. 1.3.2.1, in particular Eq. (1.40). See also [74]) the insertion of the second brane at finite position in the bulk yields a modification of the stress-energy tensor of the CFT. This leads to the conclusion that the symmetry breaking under consideration is not a spontaneous but a soft one. The generalization of the nonlocal braneworld action to one- and two-brane models with curved branes is straightforward: the major task would be to construct the form factors. For constant-curvature branes with de Sitter geometry, which are particularly interesting for considering a stage of cosmic inflation, this kernel would con- 96 4 Conclusions and Outlook

tain hypergeometric functions (cf. [134]). In principle, our method is also extendable to models with more than two branes. This would, however, lead to matrix-valued kernels of higher rank and therefore lead to increasingly complicated actions. The same is true for the inclusion of additional bulk fields. Also the method of the recovery of the KK description from the nonlocal action, described in Sec. 3.2, is universal and extendable to arbitrary two-brane models. It provides the missing link between nonlocal braneworld actions, which are especially suited for the treatment of cosmological problems [139, 45], and spectral rep- resentations of the effective action, particularly suited to particle-phenomenology considerations [68, 69]. A useful approach could be to reverse the sequence of construction of the braneworld action: recently it has been proposed that nonlocal modifications of the four- dimensional Einstein equations might provide an explanation for the observed present-day acceleration of the universe [135, 136]. Even more, nonlocal effects might provide a possibility to explain the galactic rotation curves without invoking dark matter [137]. Therefore one could try to construct a nonlocal kernel providing the desired phenomenology. Then one can infer from the structure of the kernel which braneworld setup could model our universe explaining the cosmic acceleration and the deviations from Newton’s law on galactic scales. Another aspect interesting from the viewpoint of model building would be to apply our method to the Karch-Randall (KR) one-brane model which describes the embedding of an anti-de Sitter brane in an anti-de Sitter bulk [13]. The construction of the nonlocal action for this model may lead to interesting insights into the stability of the KR model. Recently it has been shown from studies of linearized gravity that the model features a radion despite of its one-brane nature [138]. This observation might have been anticipated from studying perturbations of the brane’s worldsheet in the KR model: the equations of motion of worldsheet perturbations will not lead to an identification of the radion with the conformal mode of the induced metric because the conformal mode will not be completely localized at the brane and therefore will have a different equation of motion as the radion which is a purely four-dimensional scalar. As a result, the radion dipole-ghost term present in our one-brane action, Eq. (2.183), should not vanish on-shell in the KR model. It would therefore be interesting to study this term for the KR model in order to see if it indicates an instability or if its dipole nature is cured by some mechanism. Our result about the nature of symmetry breaking in the CFT-dual of the RS two-brane model demonstrates that the nonlocal braneworld action is also applicable to problems in the context of the AdS/CFT correspondence. This finding suggests that the nonlocal-action approach should be a convenient calculational technique for studying other aspects of the correspondence. The method of constructing the nonlocal braneworld action can be considered well understood from this work. The relation of the nonlocal method to other approaches in braneworld physics has been elaborated. Therefore this study should present a convenient starting point for the application of the calculational framework of the nonlocal braneworld action to cosmological aspects of braneworlds or theoretical considerations along the lines of the AdS/CFT correspondence. Even without such purpose-built tools as the nonlocal effective action one can still discover surprising phenomenological features of braneworld physics. In Sec. 3.3 we 4 Conclusions and Outlook 97 discussed one specific effect, radion-induced gravitational wave oscillations, which might be observable by gravitational wave interferometers. RIGO’s had escaped notion for several years, although they are clearly present already in the traditional KK description of braneworlds. Thus, one can infer that a powerful calculational method for cosmological aspects of braneworld physics can lead to important insights into the phenomenological relevance of the braneworld paradigm.

Appendix A

Anti-de Sitter space and the geometrical setting of the RS models

It is well known that Anti-de Sitter space is not a globally hyperbolic spacetime as it possesses both closed timelike curves and a timelike boundary at spatial inﬁnity (cf. [146, 147]). As both of these features are commonly considered to be unphysical, one usually restricts the physical spacetime to be only a part of the maximally extended AdS spacetime. The chosen patch will usually contain a part of AdS which gives the geodesics from the initial hypersurface just enough time to reconverge. In the paradigm of the Lorentzian AdS/CFT correspondence one usually chooses the patch of AdS which is described by Poincar´e coordinates,

2 3 2 l 2 2 2 ds = 2 dt + dxi + dz , (A.1) z − ! Xi=1 where l is the curvature radius of AdS space. This patch covers half of the maximally extended AdS space and is commonly called Poincaré Anti de-Sitter space (PAdS). For ease of comparison of the conformal structures of AdS and PAdS we display the Penrose diagrams of both spacetimes in Fig. 1 and 2. In Fig. 1, T/2 denotes the time after which all geodesics and timelike curves emerging from the same point at the initial hypersurface reconverge again for the first time. In all figures we suppress the 3 spatial dimensions represented by the coordinates xi. Thus, each point in the conformal diagrams corresponds to a flat three-dimensional slice. Although the restriction to the PAdS-half of AdS spacetime removes the problem of closed timelike curves, the restriction to PAdS does not render the spacetime completely physically viable. While one of the timelike boundaries at infinity of AdS has shrunk to a point, the other timelike boundary at infinity remains present, leading to an ill-posed Cauchy problem also for PAdS. However, all requirements for a causal evolution of the spacetime are fulfilled in the RS one-brane model because in this setup the PAdS spacetime is truncated by the brane in such a way that the timelike boundary at infinity is removed. To be more precise, the original RS model actually employs two copies of truncated PAdS

99 100 Appendix A AdS space and the setting of the RS models

t=∞ t=∞ t=∞

T/2

z- ∞ ∞ ∞ - - - t= t= t= z z 0 + + z=0 z=∞ z=0 z=∞ z=0 z=∞

Figure 1: Penrose Figure 2: Penrose Figure 3: Penrose Figure 4: Penrose diagram of AdS diagram of PAdS diagram of the RS diagram of the RS one-brane model two-brane model

which are identified along the cut, i. e. the position of the brane, via imposing a Z2 symmetry. The reflection symmetry has however no influence on the causal structure of the spacetime (cf. p. 13 footnote 6 on the effects of the Z2 symmetry). The conformal diagram is given in Fig. 3. The presence of the brane renders the extent of the z coordinate semi-compact and the loss of information to infinity or the gain of information from infinity in finite time is prevented. Therefore, the Cauchy problem for the RS spacetime becomes well-posed, and the spacetime becomes globally hyperbolic.1 The addition of the second, negative-tension, brane in the RS two-brane model does not significantly alter this picture (Fig. 4). The second brane in the small volume region of the spacetime cuts off PAdS towards spatial infinity in the other direction, rendering the spatial extent in the z direction compact. Of course, it would be desirable to derive the initial conditions for a braneworld setup from fundamental principles. Some work in this direction has been done in [50] where the “creation from nothing” by an instanton transition from the Euclidian regime to the Lorentzian spacetime was considered for the generalizations of the RS braneworlds to curved branes (cf. also the remarks at the end of Sec. 2.8.2).

1Actually also in the spacetime conﬁguration underlying the AdS/CFT correspondence the Cauchy problem is well posed because the spacetime considered is not PAdS. Far from the horizon z , where the stack of branes is located, the geometry is no longer AdS space but tends to ﬂat → ∞ Minkowski space (cf. [55]). Therefore the timelike boundary at z = 0 is not present in the geometry produced by a stack of branes at the horizon. Appendix B

Diagonalization of the kinetic and mass terms

Here we describe in detail the diagonalization of the kinetic and massive operators and the calculation of the mixing parameters from Sec. 3.2.4. The procedure of the diagonalization of the matrices D and M will include three successive operations. First we shall ﬁnd the eigenvalues λ1, λ2 of the operator D and shall construct an orthogonal matrix O, which provides the transition from an old basis to the basis of normalized eigenvectors of the operator D. Application of the transformation O to the matrix D transforms it to a diagonal matrix, whose elements coincide with the eigenvalues of D, λ 0 ODOT = 1 . (B.1) 0 λ 2 Secondly, in order to transform this matrix into the unit matrix, we act on our eigenvectors by the generalized dilatation matrix ∆,

√λ 0 ∆ = 1 . (B.2) 0 √λ 2 1 The action of the inverse operators ∆− on the matrix (B.1) naturally gives the unit matrix, 1 0 ∆ 1O D OT ∆ 1 = . (B.3) − − 0 1 The simultaneous action of the operators O and ∆ on the matrix M (3.55) transforms it into 1 T 1 M˜ = ∆− O M O ∆− , (B.4) which is still degenerate. The non-zero eigenvalue of this matrix, which gives the squared mass of the massive graviton, is equal to the trace of this matrix,

M 2 = TrM˜ . (B.5)

As the third step, in order to ﬁnally obtain an expression for the transformation + from the initial gravitational modes h and h− to the new modes hM and h0 corresponding to the massive and massless gravitons, we should use another orthogonal

101 102 Appendix B Diagonalization of the kinetic and mass terms rotation Q, diagonalizing the matrix M˜ in such a way that its diagonal elements coincide with its eigenvalues, i. e.

M 2 0 QM˜ QT = . (B.6) 0 0 Naturally, the application of the orthogonal transformation Q to the transformed kinetic matrix (B.3) does not change it because it is a unit matrix. + Thus, one can formulate the transformation between the modes h and h− and the eigenmodes of the operator (3.54) in the form

h h+ M = Q∆O . (B.7) h h 0 − The inverse transformation is h+ h = OT ∆ 1QT M . (B.8) h − h − 0 It is important to notice that the transformations (B.7) and (B.8) are not orthogonal. This is connected with the fact that we had to apply the generalized dilatation matrix ∆ to normalize properly the kinetic part of the eﬀective action. The eigenvalue equation for the kinetic matrix D deﬁned in Eq. (3.56) has the following form:

λ2 λα(a4 + 6a2 + 1) + 3α2a2(a2 + 1)2 = 0 , (B.9) − where we have introduced the abbreviation α = (1 a2)/3a2(1 + a2)2. The solutions − are α(a4 + 6a2 + 1) α√a8 + 14a4 + 1 λ = . (B.10) 1,2 2 2 Now, we substitute these eigenvalues into an eigenvector equation,

(D λ I)ψ = 0 , (B.11) − 1,2 1,2

where I is a unit matrix and ψ1,2 are eigenvectors, corresponding to the eigenvalues λ1,2, respectively. We choose the normalized eigenvectors

1 D ψ = 12 , (B.12) 1 N λ D 1 1 − 11 1 D ψ = 12 , (B.13) 2 N λ D 2 2 − 11 where Dij are the corresponding elements of the matrix D, while the normalization factors are equal to

N = D2 + (D λ )2, 1 12 11 − 1 q N = D2 + (D λ )2. (B.14) 2 12 11 − 2 q Appendix B Diagonalization of the kinetic and mass terms 103

Correspondingly, the orthogonal matrix providing the transformation from the old 1 0 basis , to the new basis ψ , ψ given by the expressions (B.12), (B.13) 0 1 1 2 has the form D12 D12 N1 N2 O =  λ D λ D  . (B.15) 1 − 11 2 − 11  N1 N2  Using the expressions for the orthogonal matrix Oand for the generalized dilatation matrix ∆ one can get the expression for the rotated matrix M˜ , substituting Eqs. (B.15) and (B.2) into Eq. (B.4),

M˜ 2 M˜ m˜

1 T 1 λ1 √λ1λ2 M˜ = ∆− OMO ∆− = β   , (B.16) M˜ m˜ m˜ 2    √λ1λ2 λ2    where a2 1 M˜ = a2D , (B.17) 12 N − N 1 2 a2(r s) r + s m˜ = − + , (B.18) N1 N2 and λ λ D D r = 1 − 2 , s = 11 − 22 . (B.19) 2 2 Now we are in a position to calculate the value of the mass of the massive graviton mode by substituting the formulas (B.16) – (B.19) together with formula (B.10) and the explicit values of the matrix elements Dij from formula (3.54) into Eq. (B.5). Straightforward but rather cumbersome calculations lead us to the simple expression 24a2(1 + a2) M 2 = . (B.20) l2(1 a2)2 − To accomplish the procedure of simultaneous diagonalization of the matrices D and M we should ﬁnd the matrix Q rotating the matrix M˜ to the diagonal form (B.6). One can ﬁnd normalized eigenvectors of the matrix M˜ by solving the equations

(M˜ M 2I)φ = 0 , − 1 M˜ φ2 = 0. (B.21) These eigenvectors have the form

1 √λ M˜ φ = 2 , 1 N √λ m˜ 1 1 √λ m˜ φ = 1 , (B.22) 2 N √λ M˜ − 2 where 2 2 N = λ2M˜ + λ1m˜ . (B.23) q 104 Appendix B Diagonalization of the kinetic and mass terms

The orthogonal matrix Q is correspondingly

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The completion of this thesis would have been hardly possible, without the aid of quite a number of persons. To them I am indebted: Prof. Claus Kiefer, for entrusting me with the task of this thesis, and providing invaluable support in mastering it. Moreover, he gave me the opportunity to enhance my knowledge by visiting conferences and other institutes and relieved me from most of a Phd. student’s usual duties — not mentioning his personal qualities making working as his student especially enjoyable. My collaborator (or better said tutor) Prof. Andrei O. Barvinsky, who generously shared his unique knowledge of nonlocal methods with me. Andrei never fell short of an answer to my questions. His exceptional knowledge has contributed much to the maturity of this thesis. My collaborator (also better said tutor) Prof. Alexander Yu. Kamenshchick, who helped me keep pace when Andrei was proceeding too fast for my grasp. Sasha’s thorough advice is reflected in large portions of the thesis. Christian Heinicke, who without hesitation took the burden of proof-reading the manuscript, and also was a willing and even interested victim to my rather inarticulate thoughts and speculations on braneworlds. Prof. Hartmann Römer who willingly took over the supervision of this thesis after Claus Kiefer left Freiburg. He also considerably sharpened my view on geometrical aspects of braneworlds. Furthermore, he maintains an enjoyable atmosphere at the Department for Field Theory of the University of Freiburg, which makes working there most convenient. Andreas Sindermann, Dr. Axel Weinkauf, from the Institute for Theoretical Physics in Cologne, as well as Peter Marquard, Dr. Cornelius Paufler, and Dr. Ste- fan Waldmann, from the Department of Field Theory at Freiburg, whose unflagging background work on the computer systems is acknowledged much too seldom. Ms. Marilyn Sobieroj, Ms. Marianne Wagner of the Institute for Applied Math- ematics in Freiburg and Ms. Michaela Richmond of the Institute for Theoretical Physics in Cologne, who all provided excellent support in administrational matters. The members of the gravity group at Cologne and of the Department of Field Theory at Freiburg for sharing their physical and mathematical knowledge with me and for lots of helpful discussions. Prof. Valeri P. Frolov, Prof. Grant J. Mathews, Prof. Hans-Peter Nilles and Dr. Sergey N. Solodukhin for influential discussions on several aspects of this work.

This work was supported by the DFG-Graduiertenkolleg Nichtlineare Diﬀerential- gleichungen: Modellierung, Theorie, Numerik, Visualisierung at the Albert-Ludwigs- Universit¨at Freiburg.