The critical tension in the Cascading DGP model
aa r X i v : . [ h e p - t h ] S e p The critical tension in the Cascading DGP model
Fulvio Sbisà , ∗ and Kazuya Koyama † Institute of Cosmology & Gravitation, University of Portsmouth,Dennis Sciama Building, Portsmouth, PO1 3FX, United Kingdom Dipartimento di Fisica dell’Università di Milano
Via Celoria 16, I-20133 Milano and
INFN, Sezione di Milano,
Via Celoria 16, I-20133 Milano
Abstract
We study the behaviour of weak gravitational fields in the 6D Cascading DGP model using a bulk-based approach. To deal with the ambiguity in the thin limit of branes of codimension higher than one,we consider a specific regularization of the internal structure of the branes where the 5D brane can beconsidered thin with respect to the 4D one. We consider the solutions corresponding to pure tensionsources on the 4D brane, and study perturbations at first order around these background solutions.We adopt a 4D scalar-vector-tensor decomposition, and focus on the scalar sector of perturbations.We show that, in a suitable 4D limit, the trace part of the 4D metric perturbations obeys a decoupledequation which suggests that it is a ghost for background tensions smaller than a critical tension,while it is a healthy field otherwise. We give a geometrical interpretation of the existence of thecritical tension and of the reason why the relevant field is a ghost or not depending on the backgroundtension. We however find a value of the critical tension which is different from the one already foundin the literature. Differently from the results in the literature, our analysis implies that, choosing thebackground tension suitably, we can construct ghost-free models for any value of the free parameters ofthe theory. We suggest that the difference lies in the procedure used to evaluate the pillbox integrationacross the codimension-2 brane. We confirm the validity of our analysis by performing numerically theintegration in a particular case where the solution inside the thick cod-2 brane is known exactly. Westress that the singular structure of the perturbation fields in the nested branes set-ups is very subtle,and that great care has to be taken when deriving the codimension-2 junction conditions. ∗ [email protected] † [email protected] Introduction
Fifteen years after its discovery [1, 2], the problem of the cosmological late time acceleration remainspuzzling. On one hand, recent cosmological observations are well fitted by Λ CDM models. On the otherhand, the best fit value for the cosmological constant Λ is dramatically different from the theoreticalestimates for the value of vacuum energy [3], and yet different from zero. Along with explaining why Λ is small compared to the theoretical predictions ( old cosmological constant problem), it is now necessaryto explain also why it is non-zero and extremely fine-tuned ( new cosmological constant problem). Anintriguing interpretation of these observations is that they may signal a breakdown of General Relativity(GR) at ultra large scales, without the need of introducing an ad-hoc dark energy component.From this point of view, a promising direction is to study theories which modify GR in the infrared,while reproducing its results at length scales where the latter is well tested (the modified gravity approach).In the past years, several proposals have been developed following this idea, including f ( R ) gravity, massivegravity and braneworld models (see [4] for a review). In particular, the latter models (see [5, 6] for earlyproposals) are appealing from the point of view of high energy physics, since the existence of branes andof extra dimensions is an essential ingredient in string theory (see for example [7]). From the point ofview of the cosmological constant problem, braneworld models with infinite volume extra dimensions canbypass Weinberg’s no-go theorem [8, 9], and more specifically (in the case of codimension higher than one)they may act as a high-pass filter on the wavelength of gravitational sources, effectively “degravitating”sources which are nearly constant with respect to a characteristic length of the model [10]. From thepoint of view of the late time acceleration of the universe, the inclusion of an induced gravity term inthe brane action (pioneered by the DGP model [11]) generically allows the existence of self-acceleratingcosmological solutions, which may be used to explain the late time acceleration in a geometrical way.The DGP model, however, does not provide a viable resolution of the acceleration problem, since itsself-accelerating solutions are observationally ruled out [12, 13, 14], and are plagued by the presence ofghosts [15, 16, 17, 18, 19, 20]. A natural idea is to consider higher codimension generalizations of the DGPmodel, since increasing the codimension should soften the tension between data and cosmological solutions[21], and these models may provide a realization of the degravitation mechanism. However, increasingthe codimension seems not to help with the ghost problem [22] (however see [23, 24]). Moreover, branesof codimension higher than one suffer from the notorious shortcomings that the thin limit of a braneis not well-defined [25], and that the brane-to-brane propagator of the gravitational field diverges whenwe send to zero the thickness of the brane [26, 27] (unless we allow for Gauss-Bonnet terms in the bulkaction [28]). An important direction to explore is to consider elaborate constructions with more than onebrane (as for example intersecting brane scenarios [29, 30]), hoping that the interplay between the branesmay provide a mechanism to get rid of the ghosts. In particular, the Cascading DGP model [31] seemsparticularly interesting since it has been claimed that the gravitational field remains finite everywhere inthe thin limit of the codimension-2 brane [31, 32] (phenomenon of gravity regularization ). Moreover, ithas been shown that in the minimal (6D) formulation of the model there exists a critical value ¯ λ c for thetension of the codimension-2 brane such that first order perturbations around pure tension backgroundscontain a ghost mode or not depending on the background tension ¯ λ [31, 33].The purpose of this paper is to understand geometrically the mechanism which is responsible for theexistence of the critical tension in the 6D Cascading DGP model. To deal with the ambiguity associatedto the thin limit, we consider a specific realization of the system where the codimension-1 brane can beconsidered thin with respect to the codimension-2 brane. To study the behaviour of gravity we use abulk-based point of view, which is geometrically more suited to the characteristics of the model. Moreprecisely, we consider (background) configurations where the source on the codimension-2 brane has the2orm of pure tension, and we study the behaviour of the gravitational field at first order in perturbationsaround these solutions. We perform a 4D scalar-vector-tensor decomposition and focus on the scalarsector, since the analysis of [33] suggests that this is the only sector which is relevant for the existence ofthe critical tension.The paper is structured as follows: in section 2 we introduce our choice regarding the regularizationof the internal structures of the branes, and we describe in detail the set-up. In section 3 we study scalarperturbations at first order around the pure tension solutions. In section 4 we derive the critical tensionand we give a geometric interpretation of its existence. We furthermore discuss the difference betweenour result and the one in the literature, and we support our analysis with a numerical check. We finallypresent some conclusions in section 5. Conventions : For metric signature, connection, covariant derivative, curvature tensors and Liederivative we follow the conventions of Misner, Thorne and Wheeler [34]. The metric signature is the“mostly plus” one, and we define symmetrization and antisymmetrization without normalization. 6D in-dices are denoted by capital letters, so run from 0 to 5; 5D indices are denoted by latin letters, and runfrom 0 to 4, while 4D indices are denoted by greek letters and run from 0 to 3. The only exception is thatthe letters i , j and k indicate 2D indices which run on the extra dimensions z and y . In general, quantitiespertaining to the cod-1 brane are denoted by a tilde ˜ , while quantities pertaining to the cod-2 braneare denoted by a superscript (4) . Abstract tensors are indicated with bold-face letters, while quantitieswhich have more than one component but are not tensors (such as coordinates n -tuples for example) areexpressed in an abstract way replacing every index with a dot. When studying perturbations, the symbol ≃ indicates usually that an equality holds at linear order. We use throughout the text the (Einstein)convention of implicit summation on repeated indices, and we will use unit of measure where the speedof light has unitary value c = 1 . The Cascading DGP model [31] is a braneworld model where a N -dimensional bulk contains a recursiveembedding of branes, each one equipped with an induced gravity term. In its minimal formulation, a 4Dbrane (which ought to describe our universe) is embedded in a 5D brane which is in turn embedded in a6D bulk, and is qualitatively described by the action S = M Z B d X √− g R + M Z C d ξ p − ˜ g ˜ R + Z C d χ q − g (4) (cid:16) M R (4) + L M (cid:17) (2.1)where B indicates the bulk, C indicates the cod-1 brane and C indicates the cod-2 brane. Here g indicates the bulk metric, ˜ g indicates the metric induced on the cod-1 brane, while g (4) indicates themetric induced on the cod-2 brane and the Lagrangian L M describes the matter localized on the cod-2brane. Concerning the coordinate systems, the bulk is parametrized by the coordinates X · = ( z, y, x µ ) ,the cod-1 brane is parametrized by the coordinates ξ · = ( ξ, ξ µ ) and the cod-2 brane is parametrized bythe coordinates χ · . Similarly to the DGP model, a Z reflection symmetry is enforced across the cod-1brane; in addition to that, in the original formulation [31] another Z reflection symmetry is imposed inthe bulk in the “parallel” direction to the cod-1 brane, so that in total the bulk enjoys a Z × Z (double)reflection symmetry (by continuity, a Z symmetry is imposed on the cod-1 brane). The theory has twofree parameters, and it is convenient to use the mass scales m ≡ M M m ≡ M M (2.2)3nd the associated length scales l ≡ /m and l ≡ /m . The analysis of [31, 32] shows that gravitybehaves in a qualitatively different way depending on the relation between m and m : if m ≫ m , weakgravity “cascades” from a 6D behaviour at very large scales to a 5D behaviour at intermediate scales to a4D behaviour at small scales, while if m ≪ m there is a direct transition from a 6D behaviour at largescales to a 4D behaviour at small scales. As in the DGP model, at small scales the tensor structure ofthe weak gravitational field is different form GR’s, so at linear level the theory does not reproduce GRresults. However, it is expected that the agreement with GR is recovered at non-linear level [32] via amultiple Vainshtein mechanism [35, 36]. See [33, 37, 38, 39, 21, 40, 41, 42, 43, 44] for other studies relatedto the Cascading DGP model. It is crucial to notice that the action (2.1) a priori does not single out a unique model if the internalstructures of the branes C and C are not specified. In fact, it is well-known that the thin limit of branesof codimension higher than one is not well defined [25]: it is expected that this property does not changeif we embed a cod-2 brane inside a cod-1 brane, since, beside the freedom to choose the cod-2 internalstructure, we now have the additional freedom to choose how the internal structures of the two (cod-1and cod-2) branes are related one to the other (see [45] for a discussion on this point). In absence of arigorous proof (on the lines of [25]) of the well-definiteness of the thin limit of the Cascading DGP model,to perform a transparent analysis we should explicitly take into account the internal structures of thebranes and their mutual relationship.An interesting choice in this sense is to consider configurations where the thickness of the cod-1 braneis much smaller than the thickness of the cod-2 brane, so that the former can be considered thin withrespect to the latter. In fact, since the thin limit of a cod-1 brane is well-defined, we can describe theseconfigurations as if the cod-1 brane were (infinitely) thin, and the cod-2 brane were “ribbon” inside thecod-1 brane. Moreover, the results of [45] imply that the thin limit of the ribbon brane is well-defined (atleast at first order in perturbations), so fixing this hierarchy permits to work with thin branes. Therefore,in the following we consider only this class of configurations, to which we refer as the nested branesrealization of the Cascading DGP model (due to the close connection with the nested branes with inducedgravity set-ups introduced in [45]). It is worthwhile to be more specific about what we mean when we say that a “ribbon” brane lies inside acod-1 brane. First of all, we assume that a 5D submanifold C (the cod-1 brane) is embedded in the 6Dbulk, and we impose a Z symmetry across the cod-1 brane. Secondly, we assume that a 4D submanifold C is embedded inside the cod-1 brane, and that matter, tension and a 4D induced gravity term areconfined inside the brane C and are localized around the brane C . We impose that a Z symmetry across C holds inside the cod-1 brane. More specifically, we distinguish between a physical (thick) cod-2 brane,inside which energy and momentum are confined (the “ribbon” cod-2 brane), and a mathematical (thin)cod-2 brane ( C ), with respect to which the Z symmetry is imposed. When the thin limit of the cod-2brane is performed, the physical brane coincides with the mathematical one. Note that, differently fromthe original formulation of the Cascading DGP model [31], we do not impose a Z × Z symmetry to holdin each of the two mirror copies which constitute the bulk. In fact, the presence of a Z symmetry insidethe cod-1 brane does not imply that a double Z symmetry holds outside of it.Following the conventions of [45], we describe the position of the cod-1 brane in the bulk by theembedding function ϕ · whose component expression is ϕ A ( ξ a ) , while we describe the position of the4mathematical) cod-2 brane inside the cod-1 brane by the embedding function ˜ α · whose expression incoordinates is ˜ α a ( χ µ ) . Composing these two embedding functions, we obtain the embedding function β · of the mathematical cod-2 brane in the bulk which in coordinated reads β A ( χ µ ) = ϕ A (cid:0) ˜ α a ( χ µ ) (cid:1) . The bulkmetric g induces on the codimension-1 brane the metric ˜ g ≡ ϕ ⋆ (cid:0) g (cid:1) , where ϕ ⋆ indicates the pullback withrespect to ϕ · , and in turn the metric ˜ g induces on the codimension-2 brane a metric g (4) ≡ ˜ α ⋆ (cid:0) ˜g (cid:1) , where ˜ α ⋆ indicates the pullback with respect to ˜ α · . The solution of equations of motion for this set-up are foundby solving the Einstein equations in the bulk G = 0 (2.3)and by imposing the Israel junction conditions [46] at the cod-1 brane M h ˜ K − ˜ K ˜ g i ± + M ˜ G = ˜ T (2.4)where ˜ K is the extrinsic curvature of the cod-1 brane ( ˜ K is its trace), ˜ G is the Einstein tensor builtfrom the cod-1 induced metric and ˜ T is the generalized energy-momentum tensor on the ribbon cod-2brane. Taking advantage of the Z symmetry which holds across the cod-1 brane, it is enough to solvethe Einstein equations only in one of the two mirror copies which constitute the bulk (henceforth, witha slight abuse of language we refer to the chosen copy as the “bulk” itself), and to impose the junctionconditions at the boundary M (cid:16) ˜ K − ˜ K ˜ g (cid:17) + M ˜ G = ˜ T (2.5)The mirror symmetry present inside the cod-1 brane is explicitly realized when we use coordinatesystems on the cod-1 brane which are Gaussian Normal with respect to the brane C . Henceforth, werefer to this class of reference systems as codimension-1 Gaussian Normal reference systems (or brieflycod-1 GNC), and we indicate quantities evaluated in this coordinate systems with an overhat ˆ . Thesereference systems are always well-defined at least in a neighbourhood of C , and are constructed startingfrom a reference system χ · on the cod-2 brane and following the geodesic of C which are normal to C .We synthetically indicate the cod-1 GN coordinates as ˆ ξ · ≡ ( ˆ ξ, χ · ) , and by construction we have that [47] ˆ g ξξ ( ˆ ξ, χ · ) = 1 ˆ g ξµ ( ˆ ξ, χ · ) = 0 (2.6)The requirement that a Z symmetry across the cod-2 brane holds inside the cod-1 brane implies that,when expressed in cod-1 GNC, the µν and ξξ components of the induced metric ˆ g (as well as of thetensors ˆ G , ˆ K and ˆ T ) are symmetric with respect to the reflection ˆ ξ → − ˆ ξ , while the ξµ components areantisymmetric. Concerning the source term on the cod-1 brane, we assume that the generalized energy-momentum tensor ˜ T present on the cod-1 brane is localized around the cod-2 brane C . By this we mean that there exists a(finite) localization length l such that in cod-1 GNC the tensor ˆ T ab ( ˆ ξ, χ · ) vanishes when it is evaluatedat a distance ˆ ξ from the cod-2 brane which is bigger than l (so the thickness of the ribbon brane is l ).More specifically, we assume the following structure for the generalized energy-momentum tensor ˆ T ab ( ˆ ξ, χ · ) = − f ( ˆ ξ ) λ δ µa δ νb γ µν ( ˆ ξ ; χ · ) + ˆ T ab ( ˆ ξ, χ · ) − f ( ˆ ξ ) δ µa δ νb M G µν ( ˆ ξ ; χ · ) (2.7) Following [45], we indicate with ˜ K the pullback on the cod-1 brane of the second fundamental form. ˆ T ab , f and f vanish for | ˆ ξ | > l . In the last expression, γ µν ( ˆ ξ ; χ · ) is a one-parameter family of(tensor) functions of the 4D coordinates χ · which coincide with the 4D components of the induced metric ˆ g ab when the latter is expressed in cod-1 GNC γ µν ( ˆ ξ ; χ · ) ≡ h ˆ g ( ˆ ξ, χ · ) i µν (2.8)The one-parameter family of tensors G µν is defined such that, for every value of ˆ ξ , G µν ( ˆ ξ ; χ · ) is the 4DEinstein tensor built from the metric γ µν ( ˆ ξ ; χ · ) . The localizing functions f and f are even functions (tocomply with the Z symmetry) which are regularized versions of the Dirac delta function, i.e. they obey Z + l − l d ˆ ξ f ( ˆ ξ ) = Z + l − l d ˆ ξ f ( ˆ ξ ) = 1 (2.9)Let us comment on each of the three contributions which constitute the generalized energy-momentumtensor. Note that γ µν ( ζ ; χ · ) is the 4D metric induced on the ˆ ξ –constant (4D) hypersurface defined by ˆ ξ = ζ . This implies that, on every ˆ ξ –constant hypersurface, the term − f λ δ µa δ νb γ µν has the form ofpure tension, where the total tension λ is distributed in the ˆ ξ direction according to the function f . Inthe thin limit (where f and f tend to the Dirac delta), this terms tends to − f ( ˆ ξ ) λ δ µa δ νb γ µν ( ˆ ξ ; χ · ) → − δ ( ˆ ξ ) λ δ µa δ νb g (4) µν ( χ · ) (2.10)which is the energy-momentum tensor correspondent to having pure tension λ on the thin cod-2 brane.Therefore, the first term in the right hand side of (2.7) describes a thick pure tension source on the ribbonbrane. The term ˆ T ab ( ˆ ξ, χ · ) instead is the energy-momentum tensor of matter present inside the ribbonbrane. To formalize the idea that momentum does not flow out of the brane, we ask that the pillboxintegration of the normal and mixed components of ˆ T ab vanishes Z + l − l d ˆ ξ ˆ T ξξ ( ˆ ξ, χ · ) = Z + l − l d ˆ ξ ˆ T ξµ ( ˆ ξ, χ · ) = 0 (2.11)and we define the cod-2 matter energy-momentum tensor as the 4D tensor T (4) µν ( χ · ) obtained by the pillboxintegration of the 4D components of ˆ T ab Z + l − l d ˆ ξ ˆ T ab ( ˆ ξ, χ · ) = δ µa δ νb T (4) µν ( χ · ) (2.12)The term − f ( ˆ ξ ) δ µa δ νb M G µν ( ˆ ξ ; χ · ) is instead a 4D induced gravity term which, instead of being localizedat ˆ ξ = 0 , is distributed in the ˆ ξ direction according to the function f . In the thin limit this term tendsto − f ( ˆ ξ ) δ µa δ νb M G µν ( ˆ ξ ; χ · ) → − δ ( ˆ ξ ) δ µa δ νb M G (4) µν ( χ · ) (2.13)which is the induced gravity term for the thin cod-2 brane. To conclude, the generalized energy-momentumtensor (2.7) corresponds to a configuration where matter, tension and a 4D induced gravity term aredistributed on a thick ribbon brane (see [32] for a discussion of the gauge invariance properties of theprocedure of smoothing a localized action). 6 Scalar perturbations around pure tension solutions
We now study the behaviour of weak gravitational fields in the nested branes realization of the 6D Cas-cading DGP model. More precisely, we consider pure tension (background) configurations, and studyperturbations at first order around these configurations. We consider a 4D scalar-vector-tensor decompo-sition of the perturbation modes, and focus on the scalar sector, which is relevant for the critical tension.We indicate the quantities correspondent to the background configurations with an overbar ¯ . Let’s consider localized source configurations where matter is absent ˜ T ab = 0 , so in cod-1 Gaussian NormalCoordinates the generalized energy-momentum tensor is of the form ¯ T ab ( ˆ ξ, χ · ) = − f ( ˆ ξ ) ¯ λ δ µa δ νb γ µν ( ˆ ξ ; χ · ) − f ( ˆ ξ ) δ µa δ νb M ¯ G µν ( ˆ ξ ; χ · ) (3.1)where ¯ λ > . It has been shown in [45] (building on the previous works [48, 49, 50]) that when M = 0 there exist solutions where the (mathematical) cod-2 brane C is placed at ξ = 0¯ α a ( χ · ) = (cid:0) , χ µ (cid:1) (3.2)and the bulk is flat ¯ g AB ( X · ) = η AB (3.3)while the cod-1 brane has the following embedding ¯ ϕ A ( ξ · ) = (cid:0) Z ( ξ ) , Y ( ξ ) , ξ µ (cid:1) (3.4)where Z ′ (cid:0) ˆ ξ (cid:1) = sin S (cid:0) ˆ ξ (cid:1) Y ′ (cid:0) ˆ ξ (cid:1) = cos S (cid:0) ˆ ξ (cid:1) (3.5)and the “slope function” S reads S ( ˆ ξ ) = ¯ λ M Z ˆ ξ f (cid:0) ζ (cid:1) dζ (3.6)Since in this case the metric γ µν ( ˆ ξ ; χ · ) is the 4D Minkowski metric, these configurations are solutions ofthe equations of motion also when M = 0 . We can freely impose the conditions Z (0) = Y (0) = 0 so that Z is even with respect to the parity transformation ˆ ξ → − ˆ ξ while Y is odd.It is easy to see [45] that, outside the thick cod-2 brane, the slope function is constant S ( ˆ ξ ) = ± S + ≡ ¯ λ M for ˆ ξ ≷ ± l (3.7)and is determined only by the total amount of tension ¯ λ (it is independent of how the tension is distributedinside the brane). Therefore, the thin limit of these solutions exists and is given by the configurationswhere the tension ¯ λ is perfectly localized on C and the components of the embedding function read Z ( ˆ ξ ) = sin (cid:16) ¯ λ M (cid:17) | ˆ ξ | Y ( ˆ ξ ) = cos (cid:16) ¯ λ M (cid:17) ˆ ξ (3.8)7ote that the normal 1-form reads ¯ n A ( ˆ ξ ) = (cid:0) Y ′ ( ˆ ξ ) , − Z ′ ( ˆ ξ ) , , , , (cid:1) (3.9)and becomes discontinuous in the thin limit. The complete 6D spacetime which corresponds to these thinlimit configurations is the product of the 4D Minkowski space and a two dimensional cone of deficit angle α = ¯ λ/M . When ¯ λ → πM − the deficit angle tends to π , and the 2D cone tends to a degenerate cone(a half-line). Therefore there is an upper bound ¯ λ < ¯ λ M ≡ πM on the tension which we can put onthe thin cod-2 brane. As we discuss in [45], to find consistent solutions in the thin limit of the ribbon brane we need eitherembedding functions which are cuspy, or a bulk metric which is discontinuous, or both. This is necessaryto produce a delta function divergence in the left hand side of the junction conditions (which balancesthe delta function divergence on the right hand side), while at the same time keeping the gravitationalfield on the thin cod-2 brane finite (gravity regularization).The choice to privilege a smooth bulk metric, without constraining the form of the embedding, or toprivilege a smooth embedding, leaving the bulk metric free to have discontinuities, is related to adoptinga bulk-based or a brane-based point of view. We suggest in [45] that the bulk-based approach have severaladvantages. In fact, it permits to identify clearly the degrees of freedom which are responsible for thesingularity (the embedding functions), separating them from the degrees of freedom which are not (thebulk metric). Moreover, the property of gravity being finite is mirrored by the fact that all the degreesof freedom remain continuous in the thin limit, so the regularity properties of the solutions are tightlylinked to the regularity properties of the gravitational field. In addition, the global geometry of the thinlimit configurations is more transparent in the bulk-based approach, for example in the pure tension casethe deficit angle is directly connected to the slope of the embedding. As we shall see, there is also amore technical reason in favour of this choice: the bulk-based approach permits us to identify clearly theconvergence properties of the perturbative degrees of freedom in the thin limit.For these reasons, we adopt the bulk-based approach to study small perturbations around the puretension solutions, following closely the analysis of [45].
We consider the following perturbative decomposition for the bulk and bending degrees of freedom g AB ( X · ) = ¯ g AB ( X · ) + h AB ( X · ) (3.10) ϕ A ( ξ · ) = ¯ ϕ A ( ξ · ) + δϕ A ( ξ · ) (3.11)and do not constrain the form of h AB and δϕ A , so we leave both the bulk metric and the cod-1 embeddingfree to fluctuate. We instead decide to keep fixed the position of the cod-2 brane inside the cod-1 brane,and we still use the 4D coordinates of the cod-1 brane to parametrize the cod-2 brane, so the embeddingof the cod-2 brane in the cod-1 brane reads ˜ α a ( χ · ) = ¯ α a ( χ · ) = (cid:0) , χ µ (cid:1) δ ˜ α a ( χ · ) = 0 (3.12) As we discuss in [45], this is the choice of the normal form with the correct orientation. β A ( χ · ) = ¯ β A ( χ · ) + δβ A ( χ · ) (3.13)where ¯ β A ( χ · ) = ¯ ϕ A (cid:0) , χ · (cid:1) δβ A ( χ · ) = δϕ A (cid:0) , χ · (cid:1) (3.14)We define the perturbations of the metric induced on the cod-1 brane as follows ˜ h ab ( ξ · ) ≡ ˜ g ab ( ξ · ) − ¯ g ab ( ξ · ) (3.15)and analogously we define the perturbation of the metric induced on the cod-2 brane as h (4) µν ( χ · ) ≡ g (4) µν ( χ · ) − ¯ g (4) µν ( χ · ) (3.16)It is useful to introduce the vectors tangent to the cod-1 brane in the ξ direction v A ≡ ∂ϕ A ∂ξ (3.17)which can be perturbatively decomposed as follows v A = ¯ v A + δv A (3.18)where ¯ v A = (cid:0) Z ′ , Y ′ , , , , (cid:1) δv A = δϕ A ′ (3.19)and we indicated a derivative with respect to ξ with a prime ′ . We adopt the convention that indiceson background quantities and on perturbations are lowered/raised with the background metric and itsinverse, so for example ¯ v A = η AL ¯ v L . It follows that the 2D indices i , j and k , which run on the extradimensions z and y , are raised/lowered with the identity matrix, so we have for example ¯ v i = ¯ ϕ ′ i ≡ δ ij ¯ ϕ j ′ ¯ n i ≡ δ ij ¯ n j (3.20)Concerning the source term, we perturb both the matter content and the tension of the cod-2 brane,so in cod-1 GNC we have ˆ T ab = ¯ T ab − δ µa δ νb f ( ˆ ξ ) ¯ λ ˆ h µν + δ ˆ T ab (3.21)where ¯ T ab is the background source term ¯ T ab = − δ µa δ νb f ( ˆ ξ ) ¯ λ η µν (3.22)and δ ˆ T ab ( ˆ ξ, χ · ) = − δ µa δ νb f ( ˆ ξ ) δλ η µν + ˆ T ab ( ˆ ξ, χ · ) − f ( ˆ ξ ) δ µa δ νb M G µν ( ˆ ξ ; χ · ) (3.23)where δλ is the perturbation of the tension. 9 .2.2 The 4D scalar-vector-tensor decomposition To deal with the issue of gauge invariance, we perform a 4D scalar-vector-tensor decomposition of the bulkand brane degrees of freedom and work with gauge invariant variables. We consider in fact the followingdecomposition of the bulk metric perturbations h µν = H µν + ∂ ( µ V ν ) + η µν π + ∂ µ ∂ ν ̟ (3.24) h zµ = A zµ + ∂ µ σ z (3.25) h yµ = A yµ + ∂ µ σ y (3.26) h yy = ψ (3.27) h zy = ρ (3.28) h zz = ω (3.29)where all these quantities are functions of the bulk coordinates X · , and we used the notation ∂ µ ≡ ∂/∂x µ .In particular, H µν is a transverse-traceless symmetric tensor while V µ , A zµ and A yµ are transverse1-forms, and ω , ρ , ψ , σ z , σ y , π and ̟ are scalars.Concerning the codimension-1 brane, we consider the scalar-vector-tensor decomposition with respectto the 4D coordinates ξ µ . Regarding the embedding, the bending modes δϕ z and δϕ y are scalars, whilethe 4D components can be decomposed as δϕ µ = δϕ T µ + ∂ ξ µ δϕ (3.30)where δϕ is a scalar and δϕ T µ is a transverse vector, and ∂ ξ µ ≡ ∂/∂ξ µ . Regarding the cod-1 inducedmetric, its decomposition is naturally linked to the decomposition of the bulk metric since each sector(scalar/vector/tensor) of the induced metric contains only the bulk perturbations of the correspondingsector [45]. This is true in turn for the decomposition with respect of the coordinates χ · of the (double)induced metric on the mathematical cod-2 brane C . To avoid a cumbersome notation, we use henceforththe convention that the evaluation on the cod-1 brane of a bulk quantity is indicated with a tilde (or withan overhat if we are using cod-1 GNC), so for example ˜ π ( ξ · ) ≡ π ( X · ) (cid:12)(cid:12)(cid:12) X · = ¯ ϕ · ( ξ · ) ˆ π ( ˆ ξ, χ · ) ≡ π ( X · ) (cid:12)(cid:12)(cid:12) X · = ¯ ϕ · (ˆ ξ · ) (3.31)and we use the convention that the evaluation on the (mathematical) cod-2 brane of a bulk quantity isindicated with a superscript (4) , so for example π (4) ( χ · ) ≡ π ( X · ) (cid:12)(cid:12)(cid:12) X · = ¯ ϕ · (0 ,χ · ) (3.32)Regarding the matter cod-1 energy-momentum tensor, we consider the following decomposition ˜ T µν = ˜ T µν + ∂ ξ ( µ ˜ B ν ) + ∂ ξ µ ∂ ξ ν ˜ T de + η µν ˜ T tr (3.33) ˜ T ξµ = ˜ D µ + ∂ ξ µ ˜ τ (3.34)where the symmetric tensor ˜ T µν is transverse and traceless, while ˜ B µ and ˜ D µ are transverse 1-forms and ˜ T ξξ , ˜ T tr , ˜ T de are scalars. We consider also the scalar-vector-tensor decomposition of the cod-2 energy-momentum tensor with respect to the coordinates χ µ T (4) µν = T (4) µν + ∂ χ ( µ B (4) ν ) + ∂ χ µ ∂ χ ν T (4) de + η µν T (4) tr (3.35)10here T (4) µν , B (4) µ , T (4) de and T (4) tr are respectively the pillbox integration of ˆ T µν , ˆ B µ , ˆ T de and ˆ T tr , whilethe pillbox integration of ˆ D µ , ˆ τ and ˆ T ξξ vanish as a consequence of (2.11). Note that the covariantconservation of the cod-2 energy momentum tensor implies (cid:3) T (4) de + T (4) tr = 0 (3.36)which in particular permits to express T (4) tr and (cid:3) T (4) de in terms of the trace T (4) ≡ η µν T (4) µν of the mattercod-2 energy-momentum tensor, namely T (4) tr = 13 T (4) (cid:3) T (4) de = − T (4) (3.37) From now on we focus only on the scalar sector. As we showed in [45], it is possible to describe in a gaugeinvariant way both the fluctuation of the bulk metric and the fluctuation of the cod-1 brane position. Inparticular, in the scalar sector there are four “metric” gauge invariant variables π gi ≡ π (3.38) h giij ≡ h ij − ∂ ( i σ j ) + ∂ i ∂ j ̟ (3.39)where h giij synthetically indicates h gizz , h gizy and h giyy , and three “brane” gauge invariant variables δϕ igi ≡ δϕ i + h σ i − ∂ i ̟ i(cid:12)(cid:12)(cid:12) X · = ¯ ϕ · ( ξ · ) (3.40) δϕ gi ≡ δϕ + 12 ˜ ̟ (3.41)where δϕ i gi synthetically indicates δϕ z gi , δϕ y gi . In addition, we can describe in a gauge-invariant way alsothe fluctuation of the position of the (mathematical) cod-2 brane in the extra dimensions, introducing thecod-2 gauge invariant bending modes δβ igi ≡ δβ i + h σ i − ∂ i ̟ i(cid:12)(cid:12)(cid:12) X · = ¯ β · ( χ · ) (3.42)The Z symmetry present inside the cod-1 brane however implies that δβ ygi vanish identically, so therelevant gauge invariant mode which describes the movement of the cod-2 brane in the bulk is the field δβ (4) ( χ · ) = δβ zgi ( χ · ) (3.43)In the M = 0 case it is moreover possible [45] to express the equations in terms of master variables[51, 52]. In fact, the bulk equations imply that the gauge invariant variables h giij can be expressed in termsof π as follows h giij = − δ ij π − (cid:3) ∂ i ∂ j π (3.44)and so the metric part of the scalar sector can be expressed in terms of the master variable π whose bulkequation is (cid:3) π = 0 (3.45)11or the brane part, it is convenient to define the normal and parallel component of the bending δϕ ⊥ andthe parallel component δϕ q δϕ ⊥ ≡ ¯ n i δϕ igi δϕ q ≡ ¯ v i δϕ igi (3.46)Since δϕ gi does not appear in the equations of motion, and δϕ q does not appear in the thin limit [45],the normal component of the bending δϕ ⊥ is the master variable for the brane perturbations in the scalarsector (in the thin limit). It is important to keep in mind that, despite the normal and parallel componentsof the bending have an intuitive geometrical meaning when the normal vector is smooth, they are not welldefined when the normal vector is discontinuous, while the z and y components of the bending remainwell-defined.These results do not change if we add the 4D induced gravity term on the ribbon brane, since we haveexplicitly G µν ( ˆ ξ ; χ · ) = − (cid:3) ˆ H µν ( ˆ ξ, χ · ) + η µν (cid:3) ˆ π ( ˆ ξ, χ · ) − ∂ χ µ ∂ χ ν ˆ π ( ˆ ξ, χ · ) (3.47)and so the scalar sector of G µν can be expressed purely in terms of ˆ π . Therefore, π and δϕ ⊥ are the (scalar)master variables of the system also in the M = 0 case. We turn now to the equations of motion for the perturbations. As we explained in [45], when we takethe thin limit of the ribbon brane the cod-1 junction conditions (2.5) split into two sets of equations: thepure codimension-1 junction conditions, which are source-free and hold for ˆ ξ = 0 , and the codimension-2 junction conditions, which are sourced and link the value of the solution at ˆ ξ = 0 − and ˆ ξ = 0 + .The addition of the 4D induced gravity term − f ( ˆ ξ ) δ µa δ νb M G µν ( ˆ ξ ; χ · ) to the cod-1 energy-momentumtensor does not spoil the derivation of the codimension-2 junction conditions, which proceeds exactly inthe same way as in [45]. Therefore, the thin limit equations of motion for the nested branes realization ofthe Cascading DGP model are obtained simply by performing the substitution T (4) µν ( χ · ) → T (4) µν ( χ · ) − M G (4) µν ( χ · ) (3.48)in the equations derived in [45]. Furthermore, in the M = 0 case a pure tension perturbation producesa perturbation of the deficit angle δα = δλ/M while leaves flat the bulk metric [45]. Since the inducedmetric on the cod-2 brane remains flat as well, these solutions are valid also in the M = 0 case. Notethat at linear order in perturbations the effect of a pure tension perturbation and of a matter perturbationare additive, so for simplicity henceforth we consider only the pure matter perturbation case δλ = 0 (withthe exception of section 4.3 and of appendix C). Performing the substitution (3.48) in the equations derived in [45], and considering only the scalar sector,the bulk equations of motion in term of gauge invariant variables read (cid:3) π = 0 (cid:3) h giij + 3 δ ij (cid:3) π + 4 ∂ i ∂ j π = 0 ( bulk ) (3.49)while the pure cod-1 junction conditions read M (cid:18) ∂ ¯ n (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) π − (cid:3) δ ˆ ϕ ⊥ (cid:19) + 32 M (cid:3) ˆ π = 0 ( pure cod-1 brane , ξξ ) (3.50)12 M (cid:18)
12 ¯ n i ¯ v j ˆ h giij + δ ˆ ϕ ′ ⊥ (cid:19) − M ˆ π ′ = 0 ( pure cod-1 brane , ξµ ) (3.51) M (cid:3) δ ˆ ϕ ⊥ + M (cid:18) −
12 ¯ v i ¯ v j (cid:3) ˆ h giij − (cid:3) ˆ π (cid:19) = 0 ( pure cod-1 brane , ∂ µ ∂ ν ) (3.52) M (cid:18) ∂ ¯ n (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) π + 12 ¯ v i ¯ v j ∂ ¯ n (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) h giij − ¯ n i ¯ v j ∂ ¯ v (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) h giij − (cid:3) δ ˆ ϕ ⊥ (cid:19) ++ M (cid:18)
12 ¯ v i ¯ v j (cid:3) ˆ h giij + 32 ˆ π ′′ + (cid:3) ˆ π (cid:19) = 0 ( pure cod-1 brane , η µν ) (3.53)Moreover, the cod-2 junction conditions read " M δ ˆ ϕ ′ ⊥ + M sin (cid:18) ¯ λ M (cid:19)(cid:16) ˆ h gizz − ˆ h giyy (cid:17) − M ˆ π ′ + = 02 M tan (cid:18) ¯ λ M (cid:19) (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + = − T (4) + M (cid:3) π (4) ( cod-2 brane ) (3.54)where the evaluation in + is a shorthand for the evaluation on the side of the thin cod-1 brane ˆ ξ = 0 + ,and we expressed (cid:3) T (4) de in terms of T (4) using the relation (3.37). Note furthermore that the second ofthe cod-2 junction conditions can be equivalently expressed in terms of the bending of the cod-2 brane asfollows M sin (cid:18) ¯ λ M (cid:19) (cid:3) δβ (4) = − T (4) + M (cid:3) π (4) (3.55)since δβ (4) and δ ˆ ϕ ⊥ (cid:12)(cid:12) + are linked by the relation δ ˆ ϕ ⊥ (cid:12)(cid:12) + = cos(¯ λ/ M ) δβ (4) .As we discuss in [45], only two of the pure cod-1 junction conditions (3.50)–(3.53) are independent ifwe take into account the second of the bulk equations (3.49). Expressing the equations in terms of themaster variables π and δ ˆ ϕ ⊥ , we obtain the following coupled system of differential equations (cid:3) π = 0 (3.56) M ∂ ¯ n (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) π + M (cid:3) ˆ π = 0 (3.57) M ˆ π ′ (cid:12)(cid:12)(cid:12) + = " M δ ˆ ϕ ′ ⊥ − M sin (cid:18) ¯ λ M (cid:19) (cid:0) ∂ z − ∂ y (cid:1) (cid:3) π (cid:12)(cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) + (3.58)and (cid:3) δ ˆ ϕ ⊥ = 12 ∂ ¯ n (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) π + 2 ∂ ξ (cid:18) ∂ ¯ n (cid:3) (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ · ) π (cid:19) (3.59) M tan (cid:18) ¯ λ M (cid:19) (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + = − T (4) + M (cid:3) π (4) (3.60)13ote that the cod-2 junction conditions (3.54) act as boundary conditions at the thin cod-2 brane for thepure cod-1 junction conditions: (3.58) acts as a boundary condition of the Neumann type for ˆ π , while(3.60) acts as boundary condition of the Dirichlet type for (cid:3) δ ˆ ϕ ⊥ .The thin limit equations (3.56)–(3.60) were derived assuming that the bulk metric converges uniformlyto a smooth limit configuration, while the embedding functions converge to a cuspy configuration. Weshowed in [45] that, in the M = 0 case, the thin limit equations are consistent for every form of sourceconfiguration; it is however easy to see that the same analysis holds also when M = 0 . This result,together with the fact that the internal structure of the cod-2 brane do not appear in the thin limitequations, implies that the thin limit of the ribbon cod-2 brane inside the (already) thin cod-1 braneis well-defined in the nested branes realization of the Cascading DGP model (at least when consideringfirst order perturbations around pure tension solutions). Moreover, since the embedding functions arecontinuous even in the thin limit and the bulk metric is smooth, the gravitational field on the cod-2 braneis finite for every form of the matter energy-momentum tensor on the cod-2 brane. This confirms thatgravity in the Cascading DGP model is regularized by the cod-1 brane with induced gravity, as anticipatedby [31, 32]. We now focus on the dynamic of the metric master variable π on the thin codimension-2 brane, wherethe critical tension is expected to emerge. Note that, if the background tension ¯ λ is non-vanishing, the cod-2 junction condition (3.60) links thevalue of the ( (cid:3) of the) normal component of the bending δ ˆ ϕ ⊥ on the side of the cod-2 brane with thevalue of the π field on the cod-2 brane, and with the trace of the matter energy-momentum tensor onthe cod-2 brane. On the other hand, the ξξ component of the pure cod-1 junction conditions (equation(3.50)) links the ( (cid:3) of the) normal component of the bending δ ˆ ϕ ⊥ to the π field on the cod-1 brane, andto the derivative of π normally to the cod-1 brane. Evaluating the latter equation on the side of the cod-2brane ( i.e. considering the ˆ ξ → + limit of the equation (3.50)), we obtain by continuity a relationshipbetween the value of (cid:3) δ ˆ ϕ ⊥ on the side of the cod-2 brane, the value of (cid:3) π on the cod-2 brane and thederivative of π normally to the cod-1 brane on the side of the cod-2 brane M ∂ ¯ n π (cid:12)(cid:12)(cid:12) + − M (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + + 32 M (cid:3) ˆ π (cid:12)(cid:12)(cid:12) + = 0 (4.1)where the latter equation contains function of the 4D variables χ · only, and we introduced the notation ∂ ¯ n π (cid:12)(cid:12)(cid:12) + = ∂ ¯ n π (cid:12)(cid:12)(cid:12) ¯ ϕ · (ˆ ξ =0 + ) (4.2)Therefore, we can then use the two equations (3.60) and (4.1) to obtain a master equation for the field π ,using the fact that by continuity of the π field we have (cid:3) π (4) = (cid:3) ˆ π (cid:12)(cid:12) + . In fact, expressing (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12) + in terms of π (4) and T (4) using the equation (3.60), and inserting the resulting relation in the equation(4.1), we get M tan (cid:18) ¯ λ M (cid:19) ∂ ¯ n π (cid:12)(cid:12)(cid:12) + + (cid:20) M tan (cid:18) ¯ λ M (cid:19) − m M (cid:21) (cid:3) ˆ π (cid:12)(cid:12)(cid:12) + = − m T (4) (4.3)14his equation is exact (at first order in perturbations), since we didn’t take any “decoupling limit” toobtain it; considering the thin limit on the cod-2 brane allowed us to find a master equation for the field π and its derivatives at the cod-2 brane.Despite the equation (4.3) involves only the value of the field π near the cod-2 brane, the presence ofthe normal derivative ∂ ¯ n π implies that to find a solution of (4.3) we have to solve the bulk equations andthe cod-1 junction conditions, or in other words we still need to solve the complete system of differentialequations for π and δ ˆ ϕ ⊥ . However, it is possible to look for an approximate description which “decouples”the dynamics on the cod-2 brane from the dynamics in the bulk and on the cod-1 brane, with the hopeto find a master equation which describes the behaviour of π on the cod-2 brane. We consider in fact thefollowing “4D limit” | m ∂ ¯ n | ≪ | (cid:3) || m ∂ ¯ n | ≪ | (cid:3) | (4.4)which implies that, in the left hand side of equation (4.3), we can neglect the first term compared to thesecond term and to the third term: we then obtain M (cid:20) − m m tan (cid:18) ¯ λ M (cid:19)(cid:21) (cid:3) π (4) = T (4) (4.5)This is the effective master equation which describes the behaviour of π field on the cod-2 brane in the 4Dlimit (note that this limit is different from the decoupling limit considered in [31, 32] which corresponds toan effective 5D description). Crucially, in this equation the numerical coefficient which multiplies (cid:3) π (4) changes sign when ¯ λ becomes equal to the critical tension ¯ λ c = 4 M arctan (cid:18) m m (cid:19) (4.6) The sign of the coefficient multiplying (cid:3) π (4) in equation (4.5) is closely related to the fact that the field π is an effective ghost or not. In fact, the equation (4.5) tells us that, in the 4D limit, the dynamics ofthe field π (4) is described by an effective 4D action which is proportional to S (2) π (4) = Z d χ (cid:20) K ∂ µ π (4) ∂ µ π (4) + π (4) T (4) (cid:21) (4.7)where we indicated K ≡ M (cid:20) − m m tan (cid:18) ¯ λ M (cid:19)(cid:21) (4.8)The effective 4D action is in general obtained by integrating out of the (quadratic approximation of the)general action (2.1) all the other fields using the bulk equations and the junction conditions, and imposingthe conditions (4.4). However, in practice we just need to determine the value of the proportionalityconstant between the true effective action and (4.7), which can be recognized from the coupling of π (4) with the matter. Expanding at quadratic order around the Minkowski spacetime the term in the generalaction which expresses the gravity-matter coupling, we get Z d χ q − g (4) L m ≃ Z d χ h (4) µν T µν (4) = Z d χ (cid:16) π (4) T (4) + H (4) µν T µν (4) (cid:17) (4.9)15hose scalar part indicates that the action (4.7) is indeed the correct 4D effective action for π (4) . Since,with our choice of the metric signature, a field which obeys an action of the form (4.7) is a ghost if K > while it is a healthy field if K < , we conclude that the field π (4) in the nested branes realization of the6D Cascading DGP model is an effective 4D ghost if the background tension is smaller than the criticaltension ¯ λ c , while it is a healthy effective 4D field if the background tension is bigger than ¯ λ c . From thepoint of view of the action, we can say that, integrating out the other fields in the scalar sector andimposing the decoupling limit, we generate a ¯ λ -dependent contribution S (2)¯ λ = Z d χ (cid:20) − M m m tan (cid:18) ¯ λ M (cid:19) ∂ µ π (4) ∂ µ π (4) (cid:21) (4.10)to the kinetic term of π (4) which is added to the 4D part of the general Lagrangian S (2)4 = Z d χ (cid:20) M ∂ µ π (4) ∂ µ π (4) + π (4) T (4) (cid:21) (4.11)and cures the ghost if ¯ λ > ¯ λ c .Note that, strictly speaking, to claim that the field π (4) is a ghost we should perform a Hamiltoniananalysis, since from a Hamiltonian point of view the system we are studying is a constrained system. Tosee why this may be relevant, consider the limit ¯ λ → of the action (4.7). Performing this limit we obtainthe action for the scalar sector of (4D) GR: if the above reasoning regarding the sign of the kinetic termwere conclusive, we should conclude that GR itself has a ghost (this is known as the conformal factorproblem in GR). However, a careful Hamiltonian analysis of GR permits to show that the constrainedstructure of the theory renders the π (4) field non-propagating, so GR is ghost-free despite the wrong signof the kinetic term of π (4) [53, 54, 24, 55, 56]. The analysis of the GR action does not extend to the 6DCascading DGP, since in the former case π (4) is the trace part of a 4D graviton while in the latter caseit is the 4D trace of a 6D graviton, and the Hamiltonian analysis should be different in the two cases.Nevertheless, we feel that the result of π (4) being a ghost when ¯ λ < ¯ λ c and healthy when ¯ λ > ¯ λ c shouldbe confirmed by a full-fledged Hamiltonian analysis. We leave this for future work. From another pointof view, it is important to remember that we obtained this result at first order in perturbations, so thepresence/absence of the ghost should be confirmed at full non-linear level. We can now understand geometrically what is the role of the background tension concerning the dynamicsof π (4) and the sign of its kinetic term, and in particular why a critical tension emerges at all. First of all,note that the 4D limit equation (4.5) for the π (4) field can be obtained directly from the the equations(3.60) and (4.1) if we neglect the term M ∂ ¯ n π in (4.1), so we can consider the following system ofequations M tan (cid:18) ¯ λ M (cid:19) (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + = M (cid:3) π (4) − T (4) (4.12) M (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + = 32 M (cid:3) ˆ π (cid:12)(cid:12)(cid:12) + (4.13)as the 4D limit of the system (3.60)-(4.1). Furthermore, it is convenient to express these equations interms of objects which have a clear geometrical meaning also in the thin limit, and in particular it is16seful to write the equation (3.60) in terms of the bending mode δβ (4) of the cod-2 brane in the bulk. Theequations (4.12)-(4.13) then read M sin (cid:18) ¯ λ M (cid:19) (cid:3) δβ (4) − M (cid:3) π (4) = −T (4) (4.14) M (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + = 32 M (cid:3) ˆ π (cid:12)(cid:12)(cid:12) + (4.15)and need to be completed with the continuity conditions δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + = cos (cid:18) ¯ λ M (cid:19) δβ (4) (4.16) (cid:3) ˆ π (cid:12)(cid:12)(cid:12) + = (cid:3) π (4) (4.17)The equation (4.16) expresses the fact that, since the components of the embedding function δϕ A arecontinuous (the cod-1 brane “does not break”), the movement of the cod-2 brane and the movement ofthe cod-1 brane near the cod-2 brane are linked. However, since δ ˆ ϕ ⊥ (cid:12)(cid:12) + is constructed from the cod-1 embedding by projecting on the normal vector, and the background normal vector depends on thebackground tension, δ ˆ ϕ ⊥ (cid:12)(cid:12) + and δβ (4) are linked in a ¯ λ -dependent way.We can interpret the system of equations (4.14)-(4.17) in the following way. The equation (4.14) tellsus that the presence of matter on the cod-2 brane (represented by T (4) ) has two effects: on one hand,it excites the metric perturbations on the cod-2 brane (represented by π (4) ) via the 4D induced gravityterm, and in a ghostly way. On the other hand, since the 4D brane is actually part of a 6D set-upand in fact embedded into a 5D cod-1 brane, T (4) excites also the movement of the cod-2 brane in thebulk (represented by δβ (4) ), this time in a healthy way. However, it does so in a ¯ λ -dependent way, andthis excitation mechanism is the more efficient the larger the background tension, while it is completelyinefficient when ¯ λ is very small. As we already mentioned, the equation (4.16) instead tells us that, sincethe cod-2 brane is embedded inside the cod-1 brane, the movement of the cod-2 brane “drags” the cod-1brane as well; therefore the matter on the cod-2 brane indirectly excites δ ˆ ϕ ⊥ (cid:12)(cid:12) + . Passing from δβ (4) to δ ˆ ϕ ⊥ (cid:12)(cid:12) + we gain an additional ¯ λ -dependence, but the sign does not change and so T (4) excites δ ˆ ϕ ⊥ (cid:12)(cid:12) + ina healthy way. In turn, considering now the equation (4.15), δ ˆ ϕ ⊥ (cid:12)(cid:12) + excites the metric perturbations(expressed by the field ˆ π ) on the cod-1 brane via the 5D induced gravity term, still in a healthy way. Bycontinuity of the ˆ π field (equation (4.17)), the perturbation of ˆ π finally induces the perturbation of π (4) ,in a healthy way.To sum up, the presence of matter on the cod-2 brane excites the field π (4) via two separate channels:it does so directly, because of the 4D induced gravity term, and indirectly via the bending of the cod-1 brane, because of the 5D induced gravity term. Furthermore, we saw above that the first channelexcites π (4) in a ghostly and ¯ λ -independent way, while the second channel excites π (4) in a healthy and ¯ λ -dependent way. The fact that the field π (4) is a ghost or not is decided by the fact that the first or thesecond channel is more efficient than the other. In particular, the existence of the critical tension is dueto the competition between these two channels, and its value corresponds to the tension where the twochannels are equally efficient. Note finally that the existence of the second channel is entirely due to thehigher dimensional structure of the theory. This is seen from the point of view of the action as the factthat the healthy part of the effective 4D kinetic term (which cures the presence of the ghost for ¯ λ > ¯ λ c )is created by integrating out the other fields in the 6D and 5D parts of the total action.17 .2 Ghost-free regions in parameters space The results of the previous section seem to be at odds with the findings of [33] (which agree with [31]),where the following kinetic part of the 4D effective action for the field π (4) was found S dRKTcod-2 = Z d χ M (cid:18) λ m M − (cid:19) π (4) (cid:3) π (4) (4.18)which has exactly the same structure of (4.7)–(4.8) although with a different value for the critical tension.To compare the two results, note first of all that our conventions differ slightly from those of [33], sincein our cod-1 junction conditions (2.5) the mass M is multiplied by 2, while it is not so in their case.Rescaling M → M in their result, to normalize the conventions, the critical tension in their case reads ¯ λ dRKT c = 8 m M (4.19)which is different from the value (4.6) we find in our analysis. Referring to the discussion in section 2.1and in [45], one may suggest that the two results differ because we are considering different realizations ofthe Cascading DGP (i.e. different choices for the internal structures), so that in truth we are consideringdifferent models. However, as we show in appendix A, there is a coordinate transformation which links ouranalysis to that of [33], and so we are indeed studying the same set-up although in different coordinatesystems (we use a bulk-based approach, while [33] uses a brane-based approach). Note that, strictlyspeaking, there is no contradiction with the estimate of [31], since that was found in the 5D decouplinglimit which applies only for small values of the ratio m /m . Instead, the result is in sharp contradictionwith [33].This difference has important consequences for the phenomenological viability of the model. Note that,when m ≪ m (i.e. M ≪ M /M , for example when the background tension is very small ¯ λ ≪ ), ourresult reads ¯ λ c ≃ m M m = 8 m M λ dRKT c (4.20)and so coincides with the result of [33]. However, when m is not negligible with respect to m the tworesults are different, and become dramatically so when m ≫ m . It is illuminating to consider the ratiobetween the critical tension and the maximum tension λ M = 2 πM which can be placed on the cod-2brane: in our case we get ¯ λ c ¯ λ M = 2 π arctan (cid:18) m m (cid:19) (4.21)while with the result of [33] we get ¯ λ dRKT c ¯ λ M = 43 π m m (4.22)It is apparent that with our result ¯ λ c remains smaller than ¯ λ M for every value of the parameters m and m , while with the result of [33] this is true only when m . m . Crucially, our result implies that, forevery value of the free parameters of the model, there is an interval of values for the background tensionsuch that π (4) is not a ghost, and so the model is phenomenologically viable. The result of [33], on theother hand, implies that the m > m region in parameters space is phenomenologically ruled out. Sincefor m ≪ m the gravitational field Cascades from 6D to 5D to 4D progressing from very large to verysmall scales, while in the m ≫ m there is a direct transition from 6D to 4D, it was concluded in [31, 33]that the latter behaviour of the gravitational field is ruled out. Our result instead implies that bothbehaviours are viable. 18 .2.1 Thin limit and pillbox integration It is therefore very important to understand which of the two results is the correct one, or at leastunderstand why two different results are obtained. As we mentioned above, we derived the criticaltension using the ξξ component of the cod-1 junction conditions (3.50) and the cod-2 junction condition(3.60), where the latter comes from the pillbox integration across the ribbon brane of the (4D trace ofthe) derivative part of the cod-1 induced gravity term (see [45] for the detailed derivations) lim l → + Z + l − l d ˆ ξ M (cid:20) −
12 ¯ v i ¯ v j (cid:3) ˆ h giij − (cid:3) ˆ π + ¯ v ′ i (cid:3) δ ˆ ϕ igi (cid:21) = M (cid:3) π (4) − T (4) (4.23)Of these two equations, the cod-2 junction condition is the most delicate to derive, since the pillboxintegration in the nested branes with induced gravity set-ups is very subtle [45]: it is quite natural toinvestigate first if the difference can be traced back to the way the pillbox integration is performed. Inthis respect, note that the result of [31, 33] were reproduced in our approach if we performed the pillboxintegration (4.23) in such a way to obtain the following change in the equation (3.60) M tan (cid:18) ¯ λ M (cid:19) (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + → ¯ λ m (cid:3) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + (4.24)To perform the pillbox integration in a rigorous way, in [45] we introduced a sequence of braneconfigurations indexed by n ∈ N where the width l [ n ] of the ribbon brane converges to zero l [ n ] → + ,while the pillbox integration of the source configurations ˆ T [ n ] ab is independent of n (which in particularmeans that f [ n ] and f [ n ] are two representations of the Dirac delta, and T [ n ] ab converges to δ ( ˆ ξ ) δ µa δ νb T (4) µν ).For each value of n , we associated to the source configuration the corresponding solution of the equationsof motion for the bending and for the metric perturbations, so to the sequence of source configurationswe associated a sequence of geometric configurations h [ n ] AB , ¯ ϕ A [ n ] and δϕ A [ n ] . The pillbox integration wastherefore defined as the limit lim n →∞ Z + l [ n ]2 − l [ n ]2 d ˆ ξ (4.25)of the relevant equations (the µν components of the cod-1 junction conditions). We do exactly thesame here, with the only difference that now we indicate the limit configurations without the ∞ sub-script/superscript (which characterize the limit configurations in [45]), so for example ¯ ϕ i [ n ] −−−→ n →∞ ¯ ϕ i δϕ i [ n ] −−−→ n →∞ δϕ i π [ n ] −−−→ n →∞ π (4.26)It is clear that the only terms which give a non-negligible contribution to the pillbox integration are thosewhich diverge on the cod-2 brane in the thin limit. Crucially, it is not possible to say a priori which termswill diverge, but the only thing we can do is to propose an ansatz for the behaviour of the perturbationfields in the thin limit, perform the pillbox integration and a posteriori verify that the ansatz is consistent.In [45], we proposed an ansatz in which the bulk metric converges to a smooth configuration, while theembedding converges to a cuspy configuration. This is consistent with the properties of the pure tension(background) solutions, which display exactly this behaviour, and is linked to the idea that the cusp of theembedding functions supports all the singularity in the geometric configuration. This is the same ansatzwe used in the present paper to derive the thin limit equations of section 3.3. We showed in [45] that thisansatz produces a consistent system of thin limit equations, so it is consistent itself. According to thisansatz, the only terms which give a non-vanishing contribution to the pillbox integration are those which19ontain fields derived twice with respect of ˆ ξ , either embedding functions (background or bending modes)or bulk metric perturbations evaluated on the cod-1 brane. On the other hand, derivatives of every orderof the bulk perturbations with respect to the bulk coordinates are smooth, and so remain bounded whenevaluated on the cod-1 brane. It follows that the only term in (4.23) which contributes is ¯ v ′ i (cid:3) δ ˆ ϕ igi , since ¯ v ′ i = ¯ ϕ ′′ i , so the cod-2 junction condition (3.60) reads lim n → + ∞ M Z + l [ n ]2 − l [ n ]2 d ˆ ξ ¯ ϕ [ n ] ′′ i (cid:3) δ ˆ ϕ i [ n ] gi = M (cid:3) π (4) − T (4) (4.27) The subtle point is the evaluation of the integral in the left hand side of the previous equation I = lim n → + ∞ Z + l [ n ]2 − l [ n ]2 d ˆ ξ ¯ ϕ [ n ] ′′ i δ ˆ ϕ i [ n ] gi (4.28)In [45], we used the relation ¯ ϕ [ n ] ′′ i δ ˆ ϕ i [ n ] gi = (cid:16) ¯ ϕ [ n ] ′ i δ ˆ ϕ i [ n ] gi (cid:17) ′ − ¯ ϕ [ n ] ′ i δ ˆ ϕ i [ n ] ′ gi (4.29)and the fact that the second term on the right hand side does not diverge in the thin limit, since it doesnot contain second derivatives, so its pillbox integration tends to zero when n → + ∞ . The integral ofthe first term in the right hand side is trivial, and we obtain I = 2 lim n → + ∞ h ¯ ϕ ′ i [ n ] δ ˆ ϕ i [ n ] gi i l [ n ]2 = 2 lim n → + ∞ h Z ′ [ n ] δ ˆ ϕ z [ n ] gi i l [ n ]2 = 2 sin (cid:18) ¯ λ M (cid:19) δβ (4) (4.30)where we used the fact that δ ˆ ϕ ygi vanishes in ˆ ξ = 0 since it is odd and continuous. We can express thisresult in terms of the normal bending δ ˆ ϕ ⊥ , since δ ˆ ϕ ⊥ (cid:12)(cid:12) + = Y ′ (cid:12)(cid:12) + δβ (4) , to obtain I = 2 tan (cid:18) ¯ λ M (cid:19) δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + (4.31)We refer to this evaluation of the integral I as “route A”. However, we may take a different route (whichwe call “route B”): in fact, using the relations (3.4), (3.5) and (3.9) we deduce that ¯ ϕ [ n ] ′′ i = S ′ [ n ] ¯ n [ n ] i (4.32)and therefore we can write the integrand of (4.28) in terms of the normal component of the bending ¯ ϕ [ n ] ′′ i δ ˆ ϕ i [ n ] gi = S ′ [ n ] δ ˆ ϕ [ n ] ⊥ (4.33)Since, by equation (3.6), S ′ [ n ] is proportional to a realization of the Dirac delta S ′ [ n ] = ¯ λ M f [ n ]1 ( ˆ ξ ) (4.34) We take the (cid:3) out of the integral, since the functions are smooth in the 4D directions also in the thin limit.
20e may be tempted to use the defining property of the Dirac delta lim n → + ∞ Z + ∞−∞ d ˆ ξ f [ n ]1 ( ˆ ξ ) F ( ˆ ξ ) = F (0) (4.35)to evaluate the integral I as follows I = ¯ λ M lim n → + ∞ Z + l [ n ]2 − l [ n ]2 d ˆ ξ f [ n ]1 ( ˆ ξ ) δ ˆ ϕ [ n ] ⊥ = ¯ λ M δ ˆ ϕ ⊥ (cid:12)(cid:12)(cid:12) + (4.36)Comparing (4.31) to (4.36), it is evident that, using the route B to perform the integral I instead of theroute A, the term tan (cid:0) ¯ λ/ M (cid:1) is substituted by ¯ λ/ M . It follows that, if we use the route B to derivethe cod-2 junction condition (3.60) instead of the route A, we indeed generate the change (4.24) in theequation (3.60) and so we obtain exactly the result of [31, 33] for the value of the critical tension.However, it is possible to see that the route B is not mathematically well justified. In fact, its centralpoint is the use of the property of the Dirac delta (4.35) where the function F ( ˆ ξ ) is F ( ˆ ξ ) = δ ˆ ϕ [ n ] ⊥ (4.37)The use of the formula (4.35) with this identification involves a subtlety, since δ ˆ ϕ [ n ] ⊥ is a sequence offunctions: the usual proof of (4.35) assumes that F is a continuous function which is independent of n .We proved in [45] that (4.35) holds even when F is replaced by a sequence F n of continuous functions,provided it converges uniformly to a continuous function F ∞ (see [57] for the standard definitions ofpointwise and uniform convergence). More precisely, in this case we have lim n → + ∞ Z + ∞−∞ d ˆ ξ f [ n ]1 ( ˆ ξ ) F [ n ] ( ˆ ξ ) = F ∞ (0) (4.38)However, there is no guarantee that this relation holds if F n converges pointwise to some function (con-tinuous or not). To understand why, we observe that the idea behind the formula (4.35) is that, since f [ n ] is peaked around ˆ ξ = 0 , it probes the function F only around ˆ ξ = 0 . If F is continuous, in the n → + ∞ limit it can be considered nearly constant in the ˆ ξ -interval where f [ n ] is peaked, and so it can be takenout of the integral. The same happens for the formula (4.38), since the behaviour of F [ n ] near ˆ ξ = 0 isunder control if F [ n ] converges uniformly to a continuous function. On the other hand, if the convergenceof F [ n ] is not uniform, F [ n ] may develop a non-trivial behaviour (for example, a peak) around ˆ ξ = 0 in the n → + ∞ limit, as much as f [ n ] does. In this case, by no means it can be considered constant and takenout of the integral, since its singular behaviour may contribute in a non-trivial way to the integral evenin the limit.This is in fact what happens in our case. Remember that, by definition, δ ˆ ϕ [ n ] ⊥ is the projection of thebending modes on the background normal vector δ ˆ ϕ [ n ] ⊥ = ¯ n [ n ] i δ ˆ ϕ i [ n ] gi = Y ′ [ n ] δ ˆ ϕ z [ n ] gi − Z ′ [ n ] δ ˆ ϕ y [ n ] gi (4.39)The bending modes δ ˆ ϕ z [ n ] gi and δ ˆ ϕ y [ n ] gi necessarily have to converge to continuous functions ( δ ˆ ϕ zgi and δ ˆ ϕ ygi ),otherwise the cod-1 brane would break into two pieces when the ribbon brane becomes thin. Actually,the functions δ ˆ ϕ y [ n ] gi and δ ˆ ϕ ygi vanish in ˆ ξ = 0 since they are continuous and odd (as a consequence of the Z symmetry present inside the cod-1 brane), and so the normal component of the bending around themathematical cod-2 brane reads δ ˆ ϕ [ n ] ⊥ ≃ Y ′ [ n ] δ ˆ ϕ z [ n ] gi for ˆ ξ ≃ (4.40)21n the other hand, the background embedding Y ′ [ n ] displays a non-trivial behaviour inside the ribbonbrane. This is easily seen in the numerical plot in figure 6 of appendix B, obtained in the case of a puretension perturbation (using the explicit realizations (4.50) and (4.51) for n = 10 , and the same valuesof the free parameters as in section 4.3.1). The crucial point for the discussion above is that Y ′ [ n ] hasa peak localized inside the ribbon brane. As a consequence of this, also the normal component of thebending displays the same peaked behaviour, which is manifest in the plot in figure 1 (obtained againfor a pure tension perturbations with the same choices of figure 6). One may suggest that this peak - - ∆j Figure 1: The normal component of the bending δ ˆ ϕ [ n ] ⊥ in the case of a pure tension perturbation. Thevertical dashed lines are the boundaries of the ribbon brane.becomes less and less important as n gets bigger, and do not contribute in the n → ∞ limit. We addressthis point in section 4.3. For the time being, we just show that the sequence of functions δ ˆ ϕ [ n ] ⊥ cannotconverge uniformly to its limit configuration, and so the route B to evaluate the pillbox integral is notmathematically justified. Note in fact that, as we show in appendix B, the sequence Y ′ [ n ] converges to the discontinuous limit configuration (B.14). This implies that also the sequence δ ˆ ϕ [ n ] ⊥ in general convergesto a discontinuous configuration. However, by hypothesis the functions δ ˆ ϕ [ n ] ⊥ are smooth for every (finite)value of n . It is a general property that, if a sequence of smooth functions converges uniformly, then thelimit configuration is (at least) continuous. This implies that the convergence of the sequence δ ˆ ϕ [ n ] ⊥ is notuniform but merely pointwise, and so the formula (4.36) is not justified. In light of the discussion above, we propose that the reason for the difference between the result of theroute A and of the route B lies in the fact that the route B does not take properly into account thesingular structure of the fields at the cod-2 brane. To support this claim, we estimate numerically thepillbox integration I = ¯ λ M lim n → + ∞ Z + l [ n ]2 − l [ n ]2 d ˆ ξ f [ n ]1 ( ˆ ξ ) δ ˆ ϕ [ n ] ⊥ (4.41)in a particular case where the solution for the bending modes and for the metric perturbations is knownexactly (also) inside the thick cod-2 brane: the pure tension perturbation case. Since the solution is22nown exactly, we don’t need to make any hypothesis on the behaviour and on the convergence propertiesof the perturbation fields: we can perform explicitly the integration in the right hand side of (4.41) forseveral values of n , and estimate the value of the limit I by studying the asymptotic behaviour at large n . It is worthwhile to point out that the system of equations (3.56)–(3.60) (and therefore the equation(4.1)) were derived assuming δλ = 0 , so the pure tension case is not directly related to the existence ofthe critical tension (a pure tension perturbation does not excite π ). Nevertheless, concerning the integral I , the behaviour of the bending modes in the pure tension case is closely related to their behaviour inthe general case T (4) µν = 0 . Therefore, establishing if the route A or the route B (or none of the two) givesthe correct result in the pure tension case gives an invaluable indication about the correct way to performthe pillbox integration in the general case.As we show in appendix C, at linear level in δλ the normal component of the bending δ ˆ ϕ [ n ] ⊥ in the puretension perturbation case reads δ ˆ ϕ [ n ] ⊥ ( ˆ ξ ) ≃ cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) δβ (4)[ n ] + δλ M cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) Z ˆ ξ dζ ǫ [ n ] ( ζ ) cos (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) ++ δλ M sin (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) Z ˆ ξ dζ ǫ [ n ] ( ζ ) sin (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) (4.42)and in particular its value on the side of the cod-2 brane reads δ ˆ ϕ [ n ] ⊥ (cid:12)(cid:12)(cid:12) l [ n ]2 ≃ cos (cid:18) ¯ λ M (cid:19) δβ (4)[ n ] + δλ M cos (cid:18) ¯ λ M (cid:19) Z l [ n ]2 dζ ǫ [ n ] ( ζ ) cos (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) ++ δλ M sin (cid:18) ¯ λ M (cid:19) Z l [ n ]2 dζ ǫ [ n ] ( ζ ) sin (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) (4.43)where δλ is the tension perturbation and the regulating function ǫ [ n ] is defined as ǫ [ n ] ( ˆ ξ ) = Z ˆ ξ f [ n ]1 ( ζ ) dζ (4.44)Note that δβ (4)[ n ] in this case is independent of the 4D coordinates, as a consequence of the translationalinvariance in the 4D directions which is enjoyed by the pure tension solutions, and so is truly a number.Our aim is then to compute numerically the integral I [ n ] = ¯ λ M Z + l [ n ]2 − l [ n ]2 d ˆ ξ f [ n ]1 ( ˆ ξ ) δ ˆ ϕ [ n ] ⊥ ( ˆ ξ ) (4.45)and to compare the result with the value A [ n ] = 2 tan (cid:18) ¯ λ M (cid:19) δ ˆ ϕ [ n ] ⊥ (cid:12)(cid:12)(cid:12) l [ n ]2 (4.46)and with the value B [ n ] = ¯ λ M δ ˆ ϕ [ n ] ⊥ (cid:12)(cid:12)(cid:12) l [ n ]2 (4.47)23he routes A and B in fact claim that the result of the pillbox integration I of (4.28) and of (4.41) isrespectively the limit for n → ∞ of the sequences A [ n ] and B [ n ] . Since I is the limit of I [ n ] for n → ∞ ,we conclude that, if the route A is correct, the sequence A [ n ] and the sequence I [ n ] have to converge for n → ∞ lim n → + ∞ I [ n ] = lim n → + ∞ A [ n ] (4.48)while, if the route B is correct, the sequence B [ n ] and the sequence I [ n ] have to converge lim n → + ∞ I [ n ] = lim n → + ∞ B [ n ] (4.49) The normal component of the bending is expressed in terms of the quantities δλ , ǫ [ n ] ( ˆ ξ ) and δβ (4) [ n ] , whichplay the role of free parameters. The first one fixes the amplitude of the tension perturbation, and isindeed a free parameter apart from the fact that it has to satisfy the condition δλ/ ¯ λ ≪ . The regulatingfunction ǫ [ n ] ( ˆ ξ ) , instead, expresses the details of the internal structure of the cod-2 brane and is thereforefixed once we choose the system we are working with. For the purpose of checking numerically the validityof route A and B, it is enough to choose a particular realization of ǫ [ n ] and f [ n ] : we use the followingrealization of the Dirac delta f [ n ]1 ( ˆ ξ ) = n π (cid:16) (cid:0) n ˆ ξ (cid:1)(cid:17) for | ˆ ξ | ≤ πn for | ˆ ξ | > πn (4.50)and the associated regulating function ǫ [ n ] ( ˆ ξ ) = π (cid:16) n ˆ ξ + sin (cid:0) n ˆ ξ (cid:1)(cid:17) for | ˆ ξ | ≤ πn ± for ˆ ξ ≷ ± πn (4.51)whose plots for n = 10 are shown respectively in figure 2 and in figure 3. Note that in this case thethickness of the (physical) cod-2 brane is l = π/n , and indeed the thin limit l → + mathematicallycorresponds to the limit n → + ∞ . It is worthwhile to point out that the explicit form (4.50) for thefunction f [ n ] is of class C on all the real axis, but its second derivative does not exist in ˆ ξ = ± π/n .However, this is not a problem for what concerns the numerical check since the latter does not involvethe derivation of the function f [ n ] but only its integration.The quantity δβ (4)[ n ] , instead, is in general determined by the equations of motion once we specifythe source configuration. However, in the pure tension case δβ (4)[ n ] is not a 4D field but a number (as wementioned above), and its value is not fixed by the equations of motion since its 4D D’Alembertian (cid:3) δβ (4)[ n ] (which is the quantity which appears in the junction conditions) vanishes identically. This is consistentwith the fact that a rigid translation of the cod-1 and cod-2 branes is a symmetry of the system, since thebulk metric is the 6D Minkowski metric. However, our aim here is to understand which route (A or B, ornone of the two) to evaluate the integral I is correct from a mathematical point of view, independentlyof the fact that the integral itself does or does not contribute to the equations of motion. Therefore, inthe particular case we are considering, δβ (4)[ n ] can be considered a free parameter as well.To test the validity of the routes A and B, it is convenient to choose the free parameters in such away that the numbers A [ n ] and B [ n ] are quite different. Taking a look at (4.46) and (4.47) it is clear24 - f Figure 2: The realization f of the Dirac delta - - - - Ε Figure 3: The regulating function ǫ ¯ λ/ M ≪ , since the difference between A [ n ] and B [ n ] increases as ¯ λ/ M increases. It is therefore useful to choose the background tension close to the maximum tension.Moreover, we should choose δβ (4)[ n ] in such a way that δ ˆ ϕ [ n ] ⊥ is not too small at the side of the ribbon brane.These considerations prompt us to choose the background tension and the tension perturbation to be ¯ λ = 34 ¯ λ M δλ M = 0 . (4.52)which is consistent with the hypothesis that the tension perturbation is small since with this choice wehave δλ/ ¯ λ ≃ . . Furthermore, for δβ (4)[ n ] we choose the value δβ (4)[ n ] = 5 . Having chosen a specific realization of the internal structure f [ n ] and ǫ [ n ] , and having fixed the freeparameters, we can evaluate numerically the integral I [ n ] for several values of n .The results of the numerical integrations are given in table 1 with significant digits, and for claritythe same results are plotted in figure 4 (note that the plot is semi-logaritmic). It is evident that thepoints corresponding to A [ n ] (squares) converge to the points corresponding to I [ n ] (circles), while thepoints corresponding to B [ n ] (diamonds) are significantly distant from the former ones. This implies n I [ n ] . . . . A [ n ] . . . . B [ n ] . . . . Table 1: Numerical results of the pillbox integration æ æ æ æà à à à ì ì ì ì n Figure 4: Plot of the numerical results of the pillbox integrationthat, at least in the pure tension perturbation case, the route B is wrong while the route A is correct. Inparticular, the pillbox integration performed following the route B gives a lower value compared to the26illbox integration performed following the route A because the route B completely misses the peak of δ ˆ ϕ [ n ] ⊥ inside the cod-2 brane (see figure 1).The same conclusion can be reached in a slightly different way, by exploiting the fact that δβ (4)[ n ] and δλ are independent parameters. In fact, both I [ n ] and δ ˆ ϕ [ n ] ⊥ are the sum of a piece multiplied by δβ (4)[ n ] (whichwe call the “bending piece”) and a piece multiplied by δλ (which we call the “tension piece”); since thesetwo parameters are independent, if one of the equations (4.48) and (4.49) is valid then it has to be validalso separately for the bending piece and for the tension piece. Note that the bending piece of δ ˆ ϕ [ n ] ⊥ (cid:12)(cid:12) l [ n ]2 reads bending (cid:20) δ ˆ ϕ [ n ] ⊥ (cid:12)(cid:12)(cid:12) l [ n ]2 (cid:21) = cos (cid:18) ¯ λ M (cid:19) (4.53)while the bending piece of I [ n ] readsbending h I [ n ] i = ¯ λ M Z + l [ n ]2 − l [ n ]2 d ˆ ξ f [ n ] ( ˆ ξ ) cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) (4.54)The latter integral can be performed exactly changing the integration variable to ζ = ǫ [ n ] ( ˆ ξ ) , to obtainbending h I [ n ] i = 2 sin (cid:18) ¯ λ M (cid:19) (4.55)Putting together the formulas (4.53) and (4.55) we reproduce exactly the result of route Abending h I [ n ] i = 2 tan (cid:18) ¯ λ M (cid:19) bending (cid:20) δ ˆ ϕ [ n ] ⊥ (cid:12)(cid:12)(cid:12) l [ n ]2 (cid:21) (4.56) The analysis of the sections 4.2 and 4.3 confirms the suggestion that the difference between the tworesults (4.6) and (4.19) comes from the pillbox integration. It is in fact very unexpected that an a prioriinnocent procedure, such as exploiting the presence of a Dirac delta in the equations, is not justified in thenested-branes with induced gravity set-ups. This is a consequence of the fact that the singular structureof the geometry is very subtle, and indirectly confirms the belief that the singular structure of branesof codimension higher than one is in general more complex than the singular structure of codimension-1branes. In fact, while in the codimension-1 case there is essentially a unique way to render the branethin, in higher codimension set-ups there is an infinite number of non-equivalent ways to do that. Forexample, in our regularization choice the cod-2 brane is “stripe-like”, so in some sense it is a codimension-1regularization of a codimension-2 object, but we could have considered as well a circular or a cross-likeregularization (which is probably best suited when the cod-2 brane lies at the intersection of two cod-1branes), or many others. Each of these regularization choices has its own peculiar singular structure, andthe equations which describe the thin limit behaviour of the system have to be derived independently foreach case taking great care of its peculiarities.This analysis puts on firm footing our derivation of the thin limit equations for the nested-branesrealization of the 6D Cascading DGP. Concerning the critical tension, it strongly supports our claim thatthe correct value of the critical tension is (4.6), and that also models where gravity displays a directtransition 6D →
4D ( m ≫ m ) are phenomenologically viable. It is interesting to speculate that also the m ≫ m region of the parameter space may still lead to a cascading behaviour. In fact, in this case thecritical tension is very close to the maximal tension (so the deficit angle is close to π ). In this regime,27he angular direction around the codimension 2 source is quasi-compactified, resulting in an intermediate5D behaviour [58, 59]. We don’t pursue this point further in this paper, but it leaves open the possibilitythat our result is compatible with the spirit of [31].It is worthwhile to note that the use of the bulk-based approach has been fundamental in our analysis toidentify clearly the difference in the singular behaviour between different perturbation fields (bending andmetric). In fact, in a brane-based approach every information is encoded in the bulk metric perturbations,and (although possible) we feel that it is less intuitive to distinguish the converge properties of the differentcomponents of the metric. Although we already showed above that to reproduce the result of [33] in ourapproach we need to use a procedure which is not mathematically justified, it is interesting to see wherethe subtlety lies directly in the approach of [33]. In their analysis, the 4D Einstein-Hilbert action receivesa contribution from the integration across the cod-2 brane of the 5D Einstein-Hilbert action and of the5D boundary effective action induced on the cod-1 brane by the 6D Einstein-Hilbert action. The 4DEinstein-Hilbert action generates the ¯ λ -independent ghostly 4D kinetic term for the field π , while the ¯ λ -dependent healthy kinetic term on the cod-2 brane is generated by the integration across the cod-2brane of the mixed term
32 ¯ N ′ y ¯ N π (cid:3) ˜ σ (4.57)where ¯ N y and ¯ N are defined in appendix A and ˜ σ is a perturbation field related to the perturbation ofthe shift vector of the cod-1 brane. Here a prime stands for derivation with respect to y , which is thecoordinate on the cod-1 brane which is normal to the cod-2 brane, and the background relation (A.4)ensures that in the thin limit ¯ N ′ y ¯ N ∼ ¯ λ M δ ( y ) (4.58)The pillbox integration in [33] is executed as follows Z
32 ¯ N ′ y ¯ N π (cid:3) ˜ σ dy ∼ Z
32 1¯
N π (cid:3) ˜ σ ¯ λ M δ ( y ) dy ∼ λ M N π (cid:3) ˜ σ (cid:12)(cid:12)(cid:12) y =0 (4.59)and upon de-mixing the fields π and ˜ σ this term gives the healthy kinetic term for π on the cod-2 brane.However, a careful analysis shows that the field ¯ N has a non-trivial behaviour near the cod-2 brane,namely it is peaked at y = 0 as much as δϕ ⊥ is in our analysis, and that also the field ˜ σ has a non-trivialbehaviour there. For the same reason explained in section 4.2.2, the properties of the Dirac delta cannotbe used in evaluating the integral (4.59), and so the derivation is not mathematically justified. In this paper we studied the behaviour of weak gravitational fields in the 6D Cascading DGP model,with the aim of understanding geometrically why a critical tension emerges in the model. We consid-ered a specific realization of the Cascading DGP, which we called the nested brane realization, wherethe codimension-1 brane can be considered thin with respect to the codimension-2 brane. We consideredsolutions which correspond to pure tension sources on the codimension-2 brane, and studied perturba-tions of the bulk geometry and of the embedding of the codimension-1 brane at first order around thesebackground solutions. We performed a 4D scalar-vector-tensor decomposition of the perturbation fields,and focused on the scalar sector, which has been shown to be the only relevant sector concerning thecritical tension. 28e showed that the master variable of the scalar sector obeys a master equation on the cod-2 branewhen the latter is thin, and that in a suitable 4D limit its dynamics on the codimension-2 brane decouplesfrom its dynamics on the codimension-1 brane and in the bulk. The decoupled equation suggests thatthe master variable is an effective 4D ghost when the background tension is smaller than a critical value ¯ λ c , while it is a healthy field otherwise. We gave a geometrical interpretation of why the value of thebackground tension influences the fact that the master variable is a ghost or not, and how the criticaltension emerges. The value of the critical tension in our analysis is however different from the valuefound in the literature. This difference has an important implication because, contrary to the claim inthe literature, our result implies that, for every value of the free parameters of the theory, there existvalues of the background tension such that the model is ghost free. In particular, also the models wherethe behaviour of gravity undergoes a direct 6D →
4D transition when we move from large to small scalesare phenomenologically viable.We identified the source of this difference with the way the singularity at the codimension-2 braneis taken care of, and in particular with the procedure used to perform the pillbox integration across thecodimension-2 brane. We showed that the result in the literature relies on the use of the propertiesof the Dirac delta which is however not justified in these set-ups, due to the subtlety of the singularstructure of the fields inside the cod-2 brane. To provide an independent test of the validity of thetwo results, we performed the pillbox integration numerically in a particular case where the solution forthe perturbation fields inside the codimension-2 brane is known exactly. The outcome agrees with ourresult, which supports our analysis and the value of the critical tension that we found. We stress thatthe existence of the induced gravity term on the codimension-1 brane is crucial to avoid ghosts. This ismanifest in our geometrical interpretation, and can be seen also by considering the limit M → at M and M fixed. In this limit, the critical tension tends to the maximum tension ¯ λ c → ¯ λ M and thereforethe presence of the ghost is inevitable. This is compatible with the findings of [22] that the 6D (cod-2)DGP model has a perturbative ghost around flat solutions. Acknowledgments
FS wishes to thank Paolo Creminelli for hospitality at the Abdus Salam’s ICTP, Trieste, Italy where partof this work has been done. FS and KK were supported by the European Research Council’s startinggrant. KK is supported by the UK Science and Technology Facilities Council grants number ST/K00090/1and ST/L005573/1. 29
The brane-based approach
In this appendix we describe the analysis in the brane-based approach performed in [33], and its relation-ship with our analysis. Note that the conventions we use in the rest of the papers does not hold in thisappendix.
A.1 Pure tension solutions from a brane-based point of view
In [33], pure tension solutions in the Cascading DGP model were derived. In those solutions, a 6Dspacetime is covered by a coordinate chart (¯ z, ¯ y, ¯ x · ) , where ¯ z is defined on (0 , + ∞ ) and (¯ y, ¯ x · ) are definedon R , and the bulk geometry is defined by the line element ds = (cid:0) β (cid:1) d ¯ z + 2 β ¯ ǫ (¯ y ) d ¯ zd ¯ y + d ¯ y + η µν d ¯ x µ d ¯ x ν (A.1)where β is a real parameter and ¯ ǫ (¯ y ) is a smooth function which is a regularized version of the “symmetric”step function. In particular, ¯ ǫ is odd with respect to the reflection ¯ y → − ¯ y and asymptotes the value ± when ¯ y → ±∞ , so the convention is different from the one used in the main text for ǫ ( y ) (which asymptotes ± / ). Let’s suppose that a (thin) cod-1 brane is placed at ¯ z = 0 , and let’s choose to parametrize itwith the bulk coordinates (¯ y, ¯ x · ) , and that a (mathematical) cod-2 brane is placed at ¯ z = ¯ y = 0 , and let’schoose to parametrize it with the bulk coordinates (¯ x · ) . It is not difficult to see that the 6D Riemanntensor built from the metric (A.1) vanishes identically, and that the induced metrics on the cod-1 andcod-2 branes are respectively the 5D and the 4D Minkowski metrics (independently of the explicit form of ¯ ǫ ), so in particular the metric (A.1) satisfies the bulk Einstein equations. However, the extrinsic geometryof the cod-1 brane is non-trivial, since we have ¯ K µν = 0 ¯ K µy = 0 ¯ K yy = − β ¯ ǫ ′ (¯ y ) q β (cid:0) − ¯ ǫ (¯ y ) (cid:1) (A.2)Suppose now that the cod-1 brane contains a pure tension source localized around the cod-2 brane, sothat the energy-momentum on the cod-1 brane is of the form ¯ T ab = − ¯ λ ¯ f (¯ y ) δ µa δ νb η µν (A.3)where ¯ f (¯ y ) is a positive, even and normalized function (so it is a regularized version of the Dirac deltafunction) which describes the details of the distribution of the tension inside a thick cod-2 brane whoseboundaries are ¯ y = ± l . The only non-trivial component of the junction conditions reads β ¯ ǫ ′ (¯ y ) q β (cid:0) − ¯ ǫ (¯ y ) (cid:1) = ¯ λ M ¯ f (¯ y ) (A.4)which in particular implies that ¯ ǫ (¯ y ) = ± for ¯ y ≷ ± l . If ¯ λ < πM the equation above admits globalsolutions, and integrating it over the interval [ − l , l ] we obtain arctan β = ¯ λ M (A.5)which implies that for | ¯ y | > l the metric (A.1) is fixed by the total amount of tension ¯ λ present inside thethick cod-2 brane, while for | ¯ y | < l the shape of ¯ ǫ (¯ y ) explicitly depends on the details of how the tension Here a prime ′ indicates a derivative with respect to ¯ y .
30s distributed inside the thick cod-2 brane. In the thin limit, defined by l → + keeping ¯ λ constant, β remains constant while ¯ f tends to a Dirac delta and ¯ ǫ tends to the (symmetric) step function.Note that in this case the embedding of cod-1 brane is straight, even in the thin limit. However, thebulk metric becomes discontinuous in the thin limit ( ¯ g zy = β ¯ ǫ (¯ y ) ): this is necessary to generate a deltafunction divergence in the extrinsic curvature (cfr. (A.2) and (A.4)), as we mentioned in section 3.2. A.2 Equivalence with the bulk-based description
The geometry of the bulk-branes system corresponding to the metric (A.1) is however not evident. Thefact that the Riemann tensor is identically vanishing in the bulk implies that (A.1) is equivalent to aportion of a 6D Minkowski space written in a non-trivial coordinate system. To have a transparent ideaof the geometry of the configuration (A.1), we can try to find a coordinate transformation which maps itinto the 6D Minkowski space. The geometrical meaning of the configuration will then be encoded in theembedding of the cod-1 brane, which after the coordinate change will be non-trivial.
A.2.1 The change of coordinates
Let’s start from the configuration (A.1) g zz = 1 + β g zy = β ¯ ǫ (¯ y ) g yy = 1 g zµ = g yµ = 0 g µν = η µν and consider the following coordinate transformation ( ⋆ ) ¯ z (ˆ z, ˆ y, ˆ x · ) = 1 p β (cid:16) ˆ z − F (ˆ y ) (cid:17) ¯ y (ˆ z, ˆ y, ˆ x · ) = ˆ y ¯ x µ (ˆ z, ˆ y, ˆ x · ) = ˆ x µ which brings the metric into the form ˆ g zz = 1 ˆ g zy = − d F d ˆ y + β ¯ ǫ (ˆ y ) p β ˆ g yy = (cid:16) d F d ˆ y (cid:17) − d F d ˆ y β ¯ ǫ (ˆ y ) p β + 1ˆ g zµ = ˆ g yµ = 0 ˆ g µν = η µν Asking that ˆ g zy = 0 amounts to impose d F d ˆ y (ˆ y ) = β ¯ ǫ (ˆ y ) p β (A.6)which in turn implies ˆ g zz = 1 ˆ g zy = 0 ˆ g yy = 1 − β ¯ ǫ (ˆ y ) p β ! ˆ g zµ = ˆ g yµ = 0 ˆ g µν = η µν Secondly, consider the following coordinate transformation ( ⋆⋆ ) ˆ z ( z, y, x · ) = z ˆ y ( z, y, x · ) = G ( y )ˆ x µ ( z, y, x · ) = x µ g zz = 1 g zy = 0 g yy = d G dy ! " − β ¯ ǫ ( G ( y )) p β ! g zµ = g yµ = 0 g µν = η µν Asking that g yy = 1 amounts to d G dy ! = 1 + β β (cid:16) − ¯ ǫ (cid:0) G ( y ) (cid:1)(cid:17) (A.7)which implies g AB = η AB (A.8)Therefore, provided that the functions F and G exist, the composition of the two coordinates changes ( ⋆ )and ( ⋆⋆ ) transforms the initial metric (A.1) into the 6D Minkowski metric. The existence of solutions ofthe differential equation (A.6) is ensured by the fact that the function ¯ ǫ , being continuous, is primitivable.Concerning the existence of the function G , note first of all that the right hand side of (A.7) never vanishes,so there are two classes of solutions characterised by the fact that d G /dy is positive or negative. Thesetwo choices for the sign of d G /dy correspond to the fact that the new “ y ” coordinate ( y ) has the same orthe opposite orientation with respect to the old “ y ” coordinate ( ¯ y ): we choose to impose that d G /dy ispositive, which means that the y coordinate has the same orientation as ¯ y . Therefore, we can rewrite theequation (A.7) as d G dy = D (cid:0) G ( y ) (cid:1) (A.9)where D (cid:0) G (cid:1) = vuut β β (cid:16) − ¯ ǫ (cid:0) G (cid:1)(cid:17) (A.10)Since both ¯ ǫ and ¯ ǫ ′ are smooth and bounded by hypothesis, the function D is (globally) Lipschitzian:therefore, the Picard-Lindelöf theorem (see for example [60]) ensures that, for each choice of the initialcondition, there exists a unique local solution to the equation (A.9). Furthermore, the fact that D issmooth and bounded both from below and from above (we have in fact ≤ D ( G ) ≤ p β ) impliesthat the local solution can be extended to a global solution. Moreover, it also implies that G ( y ) is adiffeomorphism R → R and in particular is invertible.Therefore, we can indeed find a change of coordinates which maps the metric (A.1) into the 6DMinkowski metric: in the new reference system, the geometrical meaning of the configuration is encodedin the trajectory of the cod-1 brane, which is defined by F and G . In synthesis, we have passed from atrivial embedding and a non-trivial metric to a non-trivial embedding and a trivial metric. A.2.2 The new embedding of the cod-1 brane
To find the embedding of the cod-1 brane in the new bulk reference system, note first of all that we can stillparametrize the cod-1 brane and the cod-2 brane with the “old” coordinates (¯ y, ¯ x · ) and ¯ x · . Furthermore,as a consequence of the two coordinate changes, a point (¯ z, ¯ y, ¯ x · ) = (0 , ¯ y, ¯ x · ) on the cod-1 brane is mapped32nto the point ( z, y, x · ) = ( F (¯ y ) , G − (¯ y ) , ¯ x · ) , and in particular a point (¯ z, ¯ y, ¯ x · ) = (0 , , ¯ x · ) on the cod-2brane is mapped into the point ( z, y, x · ) = ( F (0) , G − (0) , ¯ x · ) . Therefore, the embedding of the cod-1brane into the 6D Minkowski space is then ϕ A (¯ y, ¯ x · ) = (cid:0) Z (¯ y ) , Y (¯ y ) , ¯ x · (cid:1) (A.11)where Z (¯ y ) ≡ F (¯ y ) and Y (¯ y ) ≡ G − (¯ y ) . Note that, as a consequence of (A.6) and (A.7), the componentsof the embedding function Z and Y satisfy Z ′ (¯ y ) + Y ′ (¯ y ) = 1 (A.12)The components of the embedding function (A.11) are not uniquely determined by the differentialequations (A.6) and (A.7), since to determine them we need to add some initial conditions. We choose toimpose that the position of the cod-2 brane have the same bulk coordinates before and after the coordinatechanges, which means to ask that F (0) = 0 and G (0) = 0 . The non-trivial components of the embeddingfunction of the cod-1 brane are then determined by the following Cauchy problems Z ′ (¯ y ) = β ¯ ǫ (¯ y ) p β Z (0) = 0 (A.13)and Y ′ (¯ y ) = s β (cid:0) − ¯ ǫ (¯ y ) (cid:1) β Y (0) = 0 (A.14)On the other hand, the relation (A.12) implies that there exists a function S (¯ y ) such that Z ′ (¯ y ) = sin S (¯ y ) Y ′ (¯ y ) = cos S (¯ y ) (A.15)which has to be odd to respect the parity of Z ′ and Y ′ . The Cauchy problems (A.13) and (A.14) thenimply that the derivative of S reads S ′ (¯ y ) = dd ¯ y arctan (cid:18) Z ′ (¯ y ) Y ′ (¯ y ) (cid:19) = β ¯ ǫ ′ (¯ y ) q β (cid:0) − ¯ ǫ (¯ y ) (cid:1) (A.16)and so, taking into account (A.4), S is determined by the Cauchy problem S ′ (¯ y ) = ¯ λ M ¯ f (¯ y ) S (0) = 0 (A.17)Taking a look at equations (A.8), (A.11), (A.15) and (A.17) it is manifest that the pure tension solutions(A.1) in the new coordinate system are exactly the pure tension solutions we consider in our analysis(see section 3.1). This motivates our assertion that the analysis of [33] and ours study exactly the samesystem, although in different coordinate systems. 33 .2.3 Perturbations In the perturbative study of [33] a 4D scalar-vector-tensor decomposition is considered, and the fieldwhich become a ghost for small background tensions is the trace part of the µν components of the bulkmetric perturbation, which for clarity we indicate with ¯ π (¯ z, ¯ y, ¯ x · ) (in [33] it is indicated with π ). Moreprecisely, the evaluation on the cod-2 brane (which we indicate here as ¯ π (4) (¯ x · ) = ¯ π (0 , , ¯ x · ) ) of the ¯ π field is a ghost in an effective 4D description when ¯ λ < ¯ λ c , while it is healthy when ¯ λ > ¯ λ c . Since thecoordinate transformations ( ⋆ ) and ( ⋆⋆ ) leave untouched the 4D coordinates, the π field of our analysisand the ¯ π field of [33] are linked by the transformation ¯ π (cid:0) ¯ X · ( X · ) (cid:1) = π (cid:0) X · (cid:1) (A.18)where we indicated collectively ¯ X · = (¯ z, ¯ y, ¯ x · ) and X · = ( z, y, x · ) . As a consistency check, we can usethe coordinate transformation of section A.2.1 to derive the bulk equation for ¯ π , starting from the bulkequation (cid:0) ∂ z + ∂ y + ∂ µ ∂ µ (cid:1) π ( X · ) = 0 in the bulk-based description. A tedious calculation leads to theequation ¯ (cid:3) ¯ π + 1¯ N (cid:18) ∂ z ¯ π − N y ∂ ¯ z ∂ ¯ y ¯ π + (1 + β ) ∂ y ¯ π − (1 + β ) ¯ N ′ y ¯ N ∂ ¯ z ¯ π + (1 + β ) ¯ N y ¯ N ′ y ¯ N ∂ ¯ y ¯ π (cid:19) = 0 (A.19)where we used the definitions ¯ N (¯ y ) ≡ p β − β ¯ ǫ (¯ y ) and ¯ N y (¯ y ) ≡ β ¯ ǫ (¯ y ) . The equation (A.19) isexactly the bulk equation which one obtains by varying the bulk action for the ¯ π field in [33] S = Z d ¯ X M (cid:16) ¯ N ¯ π ¯ (cid:3) ¯ π − N (cid:0) L ¯ n ¯ π (cid:1) (cid:17) (A.20)where L ¯ n ≡ ∂ ¯ z − ¯ N y ∂ ¯ y .In particular, the relation (A.18) implies that the evaluation of the ¯ π field on the cod-2 brane ¯ π (4) inthe analysis of [33] is exactly equal to the π field evaluated on the cod-2 brane π (4) ( χ · ) in our analysis ¯ π (4) = π (4) (A.21)which implies that the 4D effective actions for ¯ π (4) and π (4) have to be the same. Therefore, the valuesof the critical tension obtained using the two approaches have to agree. B Thin limit of the background solutions
In this appendix we clarify the properties of the thin limit of the background configurations, which werefer to in the main text. Since these solutions have been studied in detail in [45], we refer to that paperfor a more detailed discussion while we concentrate here only on the aspects which are more relevant forthe present paper.
B.1 The sequence of background solutions
As we mention in the main text, the thin limit of the background is performed by considering a sequenceof source configurations of the form ¯ T [ n ] ab ( ˆ ξ, χ · ) = − δ µa δ νb f [ n ]1 (cid:0) ˆ ξ (cid:1) ¯ λ γ [ n ] µν ( ˆ ξ ; χ · ) − f [ n ]2 ( ˆ ξ ) δ µa δ νb M G [ n ] µν ( ˆ ξ ; χ · ) (B.1)34here f [ n ] and f [ n ] are sequences of even functions which satisfy Z + ∞−∞ f [ n ]12 (cid:0) ˆ ξ (cid:1) d ˆ ξ = 1 f [ n ]12 (cid:0) ˆ ξ (cid:1) = 0 for | ˆ ξ | ≥ l [ n ]2 (B.2)and l [ n ] is a sequence of positive numbers that converges to zero: l [ n ] → + for n → + ∞ . There existexact solutions for this class of sources [45] such that the bulk, induced and double induced metrics arerespectively the 6D, 5D and 4D Minkowski metric, while the embedding of the cod-1 brane is n -dependentand non-trivial ¯ ϕ A [ n ] ( ˆ ξ, χ · ) = (cid:0) Z [ n ] ( ˆ ξ ) , Y [ n ] ( ˆ ξ ) , χ α (cid:1) (B.3)and the cod-2 brane sits at ˆ ξ = 0 ¯ α a [ n ] ( χ · ) = (cid:0) , χ α (cid:1) (B.4)Imposing the condition Z [ n ] (0) = Y [ n ] (0) = 0 , which implies that Z [ n ] ( ˆ ξ ) is even while Y [ n ] ( ˆ ξ ) is odd, weget [45] Z [ n ] ( ˆ ξ ) = Z ˆ ξ sin (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) dζ (B.5) Y [ n ] ( ˆ ξ ) = Z ˆ ξ cos (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) dζ (B.6)where the regulating function is ǫ [ n ] ( ˆ ξ ) ≡ Z ˆ ξ f [ n ]1 ( ζ ) dζ (B.7)and in particular it vanishes in ˆ ξ = 0 and is equal to ± for ˆ ξ ≷ ± l [ n ] . B.2 Thin limit of the embedding functions
It is possible to see that, for ˆ ξ ≷ l [ n ] , we can write Z [ n ] ( ˆ ξ ) and Y [ n ] ( ˆ ξ ) as Z [ n ] ( ˆ ξ ) = sin (cid:18) ¯ λ M (cid:19) | ˆ ξ | + Z n ] Y [ n ] ( ˆ ξ ) = cos (cid:18) ¯ λ M (cid:19) ˆ ξ ± Y n ] (B.8)where [45] lim n →∞ Z n ] = lim n →∞ Y n ] = 0 (B.9)Taking the limit n → ∞ we get the thin limit configurations for the embedding functions Z ( ˆ ξ ) = sin (cid:18) ¯ λ M (cid:19) | ˆ ξ | Y ( ˆ ξ ) = cos (cid:18) ¯ λ M (cid:19) ˆ ξ (B.10)where now ˆ ξ is defined on all the real axis.Concerning the first derivative with respect to ˆ ξ of the embedding functions, for every fixed value ˆ ξ different from zero (say positive, although the case ˆ ξ < is analogous), there exists a natural number N such that, for n ≥ N , we have l [ n ] < ˆ ξ (as a consequence of l [ n ] → ) and so ǫ [ n ] ( ˆ ξ ) = 1 / . Therefore (B.5)implies Z ′ [ n ] ( ˆ ξ > −−−−−→ n → + ∞ sin (cid:18) ¯ λ M (cid:19) (B.11)35n the other hand, again using (B.5) we conclude that Z ′ [ n ] (0) = 0 independently of n , and so Z ′ [ n ] converges to Z ′ (cid:0) ˆ ξ (cid:1) = sin (cid:16) ¯ λ/ M (cid:17) for ˆ ξ > for ˆ ξ = 0 − sin (cid:16) ¯ λ/ M (cid:17) for ˆ ξ < (B.12)Analogously, (B.6) implies that Y ′ [ n ] (0) = 1 independently of n , while for every fixed ˆ ξ = 0 we have Y ′ [ n ] ( ˆ ξ = 0) −−−−−→ n → + ∞ cos (cid:18) ¯ λ M (cid:19) (B.13)and so Y ′ [ n ] converges to Y ′ (cid:0) ˆ ξ (cid:1) = ( cos (cid:16) ¯ λ/ M (cid:17) for ˆ ξ = 01 for ˆ ξ = 0 (B.14)Note that both Z ′ and Y ′ are discontinuous. This is confirmed by the numerical plots figure 5 and 6obtained in the pure tension perturbation case for n = 10 (using the explicit realizations (4.50) and (4.51),and same values for the free parameters as in section 4.3.1). As we mention in the main text, this hasimportant consequences for the convergence properties of the sequences Z ′ [ n ] and Y ′ [ n ] , since it implies thatthey cannot converge uniformly to their limit configurations. In fact, since Z ′ [ n ] and Y ′ [ n ] are smooth for n finite, if they converged uniformly to Z ′ and Y ′ then the limit functions would necessarily be (at least)continuous. In particular, the behaviour of Y ′ [ n ] is peculiar, since in the thin limit it has a removable - - Ξ- - Z ¢ Figure 5: Numerical plot of the background embedding function Z ′ [ n ] . The vertical dashed lines indicatethe boundaries of the physical cod-2 brane.discontinuity. This may suggest that, when we perform the pillbox integration, we can indeed remove iteven for n finite, justifying the use of the route B. However, from the point of view of the character ofconvergence of the sequence, the removable discontinuity has a crucial meaning: it signals that for everyvalue of n there is a peak of finite height at ˆ ξ = 0 . It is precisely this peak which is responsible for themismatch between the results of the pillbox integration using the route A and the route B, since it givesa finite contribution to the pillbox integral for every value of n .36 - Ξ Y ¢ Figure 6: Numerical plot of the background embedding function Y ′ [ n ] . The vertical dashed lines indicatethe boundaries of the physical cod-2 brane. C Thin limit of a pure tension perturbation
To study the thin limit of a pure tension perturbation, we introduce a sequence of perturbations of thecod-1 generalized energy-momentum tensor (3.23) of the form δ ˆ T [ n ] ab ( ˆ ξ, χ · ) = − f [ n ]1 ( ˆ ξ ) δλ δ µa δ νb η µν − f [ n ]2 ( ˆ ξ ) δ µa δ νb M G µν ( ˆ ξ ; χ · ) (C.1)where δλ/ ¯ λ ≪ . The solution for the metric perturbations and bending modes in this case was obtainedin [45] by solving the perturbative equations. However, the results of section 3.1 and of appendix B implythat the exact (non-perturbative) solution when the tension on the cod-2 brane is ¯ λ + δλ is of the form g [ n ] AB = η AB ˆ g [ n ] ab = η ab g (4) [ n ] µν = η µν (C.2) ϕ A ( ˆ ξ ) = (cid:16) Z [ n ] ( ˆ ξ ) , Y [ n ] ( ˆ ξ ) , , , , (cid:17) (C.3)where Z ′ [ n ] and Y ′ [ n ] are expressed in terms of the regulating function as Z ′ [ n ] ( ˆ ξ ) = sin (cid:18) ¯ λ + δλ M ǫ [ n ] ( ˆ ξ ) (cid:19) Y ′ [ n ] ( ˆ ξ ) = cos (cid:18) ¯ λ + δλ M ǫ [ n ] ( ˆ ξ ) (cid:19) (C.4)We can then derive the perturbative solution by expanding the embedding functions at first order in δλ .In fact, at first order in δλ we have Z ′ [ n ] ( ˆ ξ ) ≃ sin (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) + cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) δλ M ǫ [ n ] ( ˆ ξ ) (C.5) Y ′ [ n ] ( ˆ ξ ) ≃ cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) − sin (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) δλ M ǫ [ n ] ( ˆ ξ ) (C.6)37hile for the background embedding we have Z ′ [ n ] ( ˆ ξ ) = sin (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) (C.7) Y ′ [ n ] ( ˆ ξ ) = cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) (C.8)and so the ˆ ξ -derivative of the z and y bending modes δ ˆ ϕ z ′ [ n ] = Z ′ [ n ] − Z ′ [ n ] and δ ˆ ϕ y ′ [ n ] = Y ′ [ n ] − Y ′ [ n ] read atfirst order in δλ δ ˆ ϕ z ′ [ n ] ( ˆ ξ ) ≃ δλ M ǫ [ n ] ( ˆ ξ ) cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) (C.9) δ ˆ ϕ y ′ [ n ] ( ˆ ξ ) ≃ − δλ M ǫ [ n ] ( ˆ ξ ) sin (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) (C.10)Integrating with respect to ˆ ξ , we obtain the z and y components of the bending modes at first order in δλδ ˆ ϕ z [ n ] ( ˆ ξ ) ≃ δβ (4)[ n ] + δλ M Z ˆ ξ dζ ǫ [ n ] ( ζ ) cos (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) (C.11) δ ˆ ϕ y [ n ] ( ˆ ξ ) ≃ − δλ M Z ˆ ξ dζ ǫ [ n ] ( ζ ) sin (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) (C.12)where we used the fact that δ ˆ ϕ y [ n ] (0) vanishes as a consequence of the Z symmetry which holds inside thecod-1 brane. Finally, we can construct the normal component of the bending δ ˆ ϕ [ n ] ⊥ = Y ′ [ n ] δ ˆ ϕ z [ n ] − Z ′ [ n ] δ ˆ ϕ y [ n ] to get δ ˆ ϕ [ n ] ⊥ ( ˆ ξ ) ≃ cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) δβ (4)[ n ] + δλ M cos (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) Z ˆ ξ dζ ǫ [ n ] ( ζ ) cos (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) ++ δλ M sin (cid:18) ¯ λ M ǫ [ n ] ( ˆ ξ ) (cid:19) Z ˆ ξ dζ ǫ [ n ] ( ζ ) sin (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) (C.13)and in particular its value on the side of the cod-2 brane reads δ ˆ ϕ [ n ] ⊥ (cid:12)(cid:12)(cid:12) l [ n ]2 ≃ cos (cid:18) ¯ λ M (cid:19) δβ (4)[ n ] + δλ M cos (cid:18) ¯ λ M (cid:19) Z l [ n ]2 dζ ǫ [ n ] ( ζ ) cos (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) ++ δλ M sin (cid:18) ¯ λ M (cid:19) Z l [ n ]2 dζ ǫ [ n ] ( ζ ) sin (cid:18) ¯ λ M ǫ [ n ] ( ζ ) (cid:19) (C.14)Note that, once we specify the internal structure f [ n ] , the solution for the bending modes (and triviallyfor the metric) is known explicitly both inside and outside the thick cod-2 brane.38 eferences [1] A. G. Riess et al, Observational evidence from supernovae for an accelerating universe and a cosmo-logical constant , Astron. J. 116: 1009–1038 (1998); ArXiv: 9805201 [astro-ph].[2] S. Perlmutter et al,
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