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Gauge-invariant fluctuations of scalar branes

A generalization of the Bardeen formalism to the case of warped geometries is presented. The system determining the gauge-invariant fluctuations of the metric induced by the scalar fluctuations of the brane is reduced to a set of Schr\"odinger-like equations for the Bardeen potentials and for the canonical normal modes of the scalar-tensor action. Scalar, vector and tensor modes of the geometry are classified according to four-dimensional Lorentz transformations. While the tensor modes of the geometry live on the brane determining the corrections to Newton law, the scalar and and vector fluctuations exhibit non normalizable zero modes and are, consequently, not localized on the brane. The spectrum of the massive modes of the fluctuations is analyzed using supersymmetric quantum mechanics.


I. INTRODUCTION
Higher dimensional topological defects allow to investigate the properties and implications of warped geometries [1][2][3]. Fields of various spin can be localized on warped backgrounds. For instance, chiral fermions can be localized on a five-dimensional domain-wall solution [4]. Moreover, also gravitational interactions can be localized on five-dimensional walls [5,6]. For a recent review see [7].
Suppose that a smooth domain-wall solution is used in order to localize fields. Such a geometry may be generated, for instance, by a scalar potential V (ϕ) with spontaneous breaking of ϕ → −ϕ symmetry [4]. Consider first the case where gravity is absent. In this case the scalar field has a normalizable zero mode [4].
If gravity is consistently included the fluctuations of the scalar wall will induce fluctuations in the metric. It is then natural to ask if all the metric fluctuations are localized on the wall. There are two problems in this analysis. The first one is that metric fluctuations are coupled to the fluctuations of the wall. The second one is that the coupled system of equations depends on the coordinate system. In order to discuss the metric fluctuations of scalar-tensor actions, the Bardeen formalism [8] has been very useful in four-dimensional backgrounds. The main idea is to parameterize the metric fluctuations by defining suitable gauge-invariant variables which do not change for infinitesimal coordinate transformations.
The Bardeen formalism is also rather effective in order to identify the coordinate systems where the gauge-invariant variables take a simple form. Having generalized the Bardeen formalism to the case of static five-dimensional warped geometries, the localization properties of metric fluctuations can be investigated. The standard lore is that if the zero mode of a given fluctuation is not normalizable, then it will not be localized on the brane and it will not affect the four-dimensional physics.
Explicit physical (thick) brane solutions have been derived in different numbers of transverse dimensions. In five dimensions physical branes can be obtained using a scalar domainwall where the scalar field depends upon the bulk radius [9][10][11][12]. In [13,14] the stability of scalar domain walls (inspired by gauged supergravity theories) has been analyzed. In more than five dimensions, physical brane solutions including also background gauge fields have been recently analyzed [16]. Singular (scalar) brane solutions have been also analyzed from different points of view [15]. The present analysis deals with the case of one transverse (bulk) coordinate. The proposed formalism can be however generalized to the case of higher dimensional transverse space.
The generalization of the Bardeen formalism to the case of compact extra-dimensions has been addressed in a cosmological context. The problem was to discuss the evolution of metric fluctuations in cosmological models with compact extra-dimensions with [17,18] and without [19] bulk scalars. In the present analysis the symmetry of the four-dimensional world is Lorentz invariance which will be used in order to classify the fluctuations of the metric [20]. The approach discussed in the present paper is then different from the ones previously discussed, for cosmological applications, in [17][18][19] and in [21].
The plan of this paper is the following. In Section II scalar-tensor models of physical branes will be briefly discussed. Emphasis will be given to the aspects relevant in the analysis of the perturbations. In Section III the Bardeen formalism will be generalized to the case of warped metrics. Variables invariant under infinitesimal coordinate transformations will be discussed. Physical gauge choices will be proposed. In Section IV the evolution for the scalar, tensor and vector modes of the geometry will be derived. The system will be reduced to a set of (decoupled) second order differential equations. In Section V the zero-modes of the gauge-invariant variables will be discussed. General conditions on the normalizability of the gravi-photon and gravi-scalar will be derived. Section VI contains some concluding remarks. In the Appendix useful technical results have been collected.

II. BACKGROUND MODELS
In five dimensions only one bulk coordinate is present and smooth domain wall configurations can be found using the following scalar-tensor action: whose related equations of motion are Notice that κ = 8πG 5 = 8π/M 3 5 . In order to facilitate the treatment of the fluctuations it is useful to write the five-dimensional metric in conformal coordinates, namely 1 : Choosing now natural gravitational units 2κ = 1, the explicit form of Eqs. (2.2)-(2.3) is:

6)
Eqs. (2.9) and (2.10) will allow to simplify some of the equations for the fluctuations.
Using this method of integration various exact solutions can be discussed [9][10][11][12]. Most of the results reported in this paper will be general but in order to fix the ideas it is useful to have in mind the following regular geometry : where b is a real number. Using Eq. (2.11) in Eq. (2.9) and integrating once the scalar field corresponding to the geometry (2.11) is Inserting w(ϕ) into Eq. (2.10) (2.13) Eqs. (2.11)-(2.13) describe a thick domain wall solution. The metric associated with this field configuration is non-singular (i.e. no poles are present in the curvature invariants). In the non-conformal coordinate system the metric is determined recalling that a(w)dw = dy.
Integrating once Eq. (2.11) a(y) = 1 cosh by . (2.14) Most of the results reported in the following sections will also hold for singular domain-wall solutions [15] where the scalar field is coupled to the action of a thin wall present for some finite value of the bulk coordinate.

A. Preliminaries
The fluctuations of the metric (2.4) are coupled to the fluctuations of the bulk scalar ϕ, as it can be appreciated by writing down the equations for the fluctuations in their general form. Contracting once Eq. (2.2) the background equations can be written as leading to the evolution equations of the fluctuations The metric and the scalar field have been separated into their background and perturbation parts: ϕ(x µ , w) = ϕ(w) + χ(x µ , w).
where the metric has been assumed in its conformally flat parametrization as in Eq. (2.4).
The metric fluctuation δG AB contains scalar, vector and tensor modes, namely, Since the four-dimensional metric is Lorentz invariant, the scalar, vector and tensor modes of the higher dimensional geometry will be classified according to Lorentz transformations as previously discussed [20] in a different context 2 . The simultaneous use of gauge-invariance and Lorentz invariance is one of the key points of the present analysis.

B. Gauge-invariant variables
The Bardeen formalism for metric fluctuations [8] will now be generalized to the case of higher-dimensional metrics containing non compact transverse dimensions of the type of the one given in Eq. (2.4). The fluctuation of the metric can be parametrized as where the Lorentz indices run over the four space-time dimensions. The tensor h µν is traceless and divergence-less corresponding to five independent components. The vectors D µ and f µ are both divergence- so that they will have, together, six independent components. The scalars E, ψ, C and ξ lead to four independent components. For infinitesimal coordinate transformations the fifteen independent metric fluctuations change as where, for the background metric of Eq. (2.4), ǫ A = a 2 (w)(ǫ µ , −ǫ w ). In Eq. (3.11) the Lie the covariant derivatives are compuetd using the background metric.
The ǫ µ can be written as the derivative of a scalar plus a vector where ∂ µ ζ µ = 0.
Using Eqs. whereas the vector transform asf Finally the scalars will transform as The prime denotes, as usual, derivation with respect to the bulk coordinate.
Gauge-invariant fluctuations corresponding to scalar and vector modes of a given geometry can be constructed as it was noticed by Bardeen [8] in a four-dimensional context and later generalized to higher (compact) dimensions [17][18][19]. Since the geometry contains two divergence-less vectors and one gauge function ζ µ , only one gauge-invariant variable is allowed. A possible choice of gauge-invariant vector fluctuation is Using Eqs. (3.14) and (3.15), it is easy to see thatṼ µ = V µ .
Since there are four scalar fluctuations in the metric and two gauge functions, i.e. ǫ and ǫ w , two gauge invariant variables can be defined. The gauge-invariant scalar fluctuations can be parametrized asΨ The gauge-invariant scalar field fluctuation will bẽ

C. Gauge choices
It is useful to fix completely the gauge freedom. Not every gauge fixes completely the coordinate system. In the following, two useful gauge choices will be discussed. If the gauge is completely fixed, five of the fifteen degrees of freedom appearing in the perturbed metric can be eliminated. Once the fluctuations are discussed in a given gauge, the gauge invariant variables can be computed in that specific gauge.
In order to fix completely the coordinate system, ǫ, ǫ w and ζ µ should be fixed. Two gauge choices fixing completely the coordinate system will now be discussed. The first one The gauge is called off-diagonal since the scalar contribution of the off-diagonal entry of the perturbed metric, i.e. ∂ µ C is non vanishing.
Another interesting gauge choice is the generalization is the longitudinal gauge where all the off-diagonal (scalar) components of the perturbed metric are vanishing: As before, using Eqs.
Not all the gauge choices have the feature of fixing completely the coordinate system.
As an example, consider the gaugẽ Using the conditions of Eqs.
Hence, even after the gauge choice (3.29) has been imposed, ǫ and ζ µ can still be shifted by an arbitrary function depending upon x ν . In this sense the gauge-fixing described by Eqs. (3.29) is not complete. Further discussions concerning gauge choices in the specific framework of [5,6] are contained in [22].

IV. EVOLUTION EQUATIONS FOR THE METRIC FLUCTUATIONS
The evolution equations of the fluctuations will now be derived. The coupled system of 3) will be diagonalized and reduced to a set of (decoupled) second order differential equations. Particular attention will be paid to the evolution of the gauge-invariant variables whose equations of motion should coincide in different gauges. This useful feature of the formalism will be used in order to cross-check the consistency of the resulting equations which will be independently derived in different gauges.

A. Perturbations in the longitudinal gauge
In the longitudinal gauge 3 the Bardeen potentials Ψ and Ξ coincide, respectively, with ψ and ξ. This simple fact can be easily appreciated by inserting the gauge fixing of Eqs.
Eq. (4.1) can be immediately reduced to a Schrödinger-like form by defining a 3/2 h µν = µ µν Eqs. (4.7)-(4.9) are crucial in order to diagonalize the full system. The constraints (4.7) and and Eq. (4.4) the following equation can be obtained: In order to get to Eq. (4.10), the background relations (2.9)-(2.10) have been used.
Defining the rescaled variable from Eq. (4.12) the following equation can be obtained (see Appendix B): is an interesting function which controls the localization properties of the scalar zero modes as it will be discussed in Section V. An equation analogous to (4.12) holds also for the other Bardeen potential, namely Ξ, thanks to the constraint of Eq. (4.7).
The canonical variable G is gauge-invariant. Recall in fact that under infinitesimal gauge Using Eq. (4.28), the constraint given in Eq. (4.30) can be expressed as Inserting Eq. (4.32) into Eq. (4.31) Finally, using the constraint provided by Eq. (4.32) in order to eliminate χ the following decoupled equation can be obtained:    Hence, gauge-invariance suggests that also in the off-diagonal gauge the equation for G will be given by Eq. (4.18). In fact, the constraints (4.28) and (4.30) can be used into Eq. (4.27) in order to eliminate the dependence on ξ and C. The resulting equation for a 3/2 χ is exactly Eq. (4.18) once we recall that, from Eq. (2.5), the double derivative of the potential with respect to ϕ can be expressed as The normal modes for tensors turn out to be for each of the polarizations where µ = a 3/2 h and η αβ is the Minkowski metric.
Fixing the longitudinal gauge the normal modes for the vector action will be where D µ = a 3/2 D µ . The normal modes for the scalar action will be where G is the variable previously defined and z = a 3/2 ϕ ′ /H. In order to get these normal modes the longitudinal gauge has been selected. The calculation can be also done without assuming any specific gauge (as discussed in [17], in a different context). The advantage of working with the full (i.e. non gauge -fixed) action is that the constraints will naturally appear.
Suppose, for instance, that the action of Eq. (2.1) is perturbed to second order in the amplitude of the vector fluctuations without fixing the specific gauge. Hence, in the perturbed action, after integration by parts and up to total derivative terms, a term like will appear on top of the kinetic term for D µ . By taking the derivative of the perturbed action with respect to ∂ µ f ν the constraint of Eq. The evolution equation of each tensor polarization can be written as where sh = µ and with s = a 3/2 . The zero mode of this equation corresponds to where K is a constant which should be fixed by normalization. The normalization condition The zero mode is then normalized provided a 3 (w) goes to zero to infinity faster than 1/w and provided a(w) is regular in all the domain of definition of w. Suppose, for, instance, that a(w) = (b 2 w 2 + 1) −1/2 as obtained, in a particular case, in Eq. (2.11). In this case The massive modes of Eq. (5.1) obey a Schrödinger-like equation which can be written where the effective "potential" term is In Eq. (5.6) J is the superpotential usually defined in the context of supersymmetric quantum mechanics [24]. Using supersymmetric quantum mechanics Eq. (5.5) can be written as A † Aµ m = m 2 µ m ,

C. Localization of the scalar zero modes
The evolution equation for the Bardeen potentials can be written as The same equation holds also for the rescaled Ξ. From Eq. (5.13) the zero mode is given by The normalization condition is given by, which implies that the scalar zero mode is normalizable provided 1/z 2 (w) goes to zero for |w| → ∞ faster than 1/w, i.e.
and provided z(w) is everywhere regular. In the background given by Eqs. (2.11)-(2.12) the scalar zero mode is not normalizable. In fact the normalizability condition requires that is finite. This does not happen since the integrand is not convergent for |w| → ∞. Since Eq. (5.13) is second order there will be also a second (linearly independent) solution to Eq. (5.13) for the zero mode, namely Φ 0 (w) ∼ z −1 (w) w z 2 (w ′ )dw ′ . If the first solution (going as z −1 (w)) is not normalizable, also the second one will not be normalizable [23]. For instance, in the case of the solution (2.11)-(2.13) the second zero mode blows up, for large |w| as |w| 5/2 .
The massive modes of Φ will obey a Schrödinger-like equation whose associated effective potentials and superpotentials are The equation for the canonical normal mode G reads which implies that the zero mode should go as G(w) = Kz(w). This implies that the normalization condition is This normalization integral implies that z(w) has to be always finite for any w and that it should go to zero, for large |w|, faster than 1/w. Since the integrand appearing in Eq.
which is not convergent for |w| → 0.
The massive modes related to G can be, again, described by a Schrödinger-like equation of the type of the one reported in Eq. (5.17). Notice that ther is an interesting "duality" relation among the effective potentials. The effective potential for the equation for G is In terms of the superpotential this symmetry implies that for z → 1/z The two Schrödinger-like equations obeyed by the massive modes of G and Φ can then be written, in terms of the operators defined in Eq. (5.8) as with J = z ′ /z. In the context of supersymmetric quantum mechanics these two potentials V Φ and V G are supersymmetric partners potentials [24]. This property implies that the spectra are related.

VI. CONCLUDING REMARKS
If physical walls in five dimensions are described using Eqs. (2.5)-(2.7), then in order to have localized zero modes of the metric the following integrals should be simultaneously convergent: , (6.2) 3) Using the explicit form of z(w), and recalling that, from Eq. (2.10), ϕ ′2 = 6(H 2 − H ′ ), Eq. (6.3) can be written as dw. Eq. (6.4) can be written as If I tens is convergent at infinity, then I G will be convergent in the same limit but this does not imply the localization as discussed in Section V.
The integral relations obtained in the present investigation are of general relevance for warped scalar-tensor backgrounds and they can be exploited in order to check for the normalizability of the scalar and vector zero modes of the geometry. Depending upon the specific model, the range of variation of w can be changed. However, the localization of the zero modes will always be conrtoled by the functions derived in the present analysis.
In this paper, the Bardeen formalism has been extended to the case of warped metrics and the localization of scalar and vector modes of the geometry has been investigated. All The results of the present investigation can be generalized to higher dimensions. Of particular interest, in this framework, are six-dimensional models which are relevant both in the context of compact [25][26][27] and non compact [28][29][30][31][32][33] extra-dimensions. Six-dimensional (non-singular) brane solutions have been recently derived in the context of the Abelian-Higgs model [16] and their fluctuations can be investigated with suitable extensions of the techniques discussed in the present analysis.

ACKNOWLEDGMENTS
The author is deeply indebted to M. E. Shaposhnikov for inspiring discussions and valuable comments. The author wishes also to thank G. Veneziano and M. Gasperini for their kind interest in the subject of the present analysis.

OFF-DIAGONAL GAUGES
In the longitudinal gauge the fluctuation of the metric is: δG AB = a 2 (w) 2h µν + 2η µν ψ D µ D µ 2ξ . (A.1) The Christoffel connections perturbed to first order in δG AB are Using Eqs. (A.2) the Ricci tensors perturbed to first order in the metric fluctuations can be computed with some trivial algebra involving the use of the Palatini identities: where the background value of the Christoffel connections has been used.
In the off-diagonal gauge the perturbed metric takes the form δG AB = a 2 (w) . (A.5) Following the same steps outlined in the discussion of the longitudinal gauge, the fluctuation of the Christoffel connection can be easily obtained From Eq. (A.6) the fluctuations of the Ricci tensors are, after some algebra, To eliminate the first derivative term is trivial since it suffice to redefine Ψ as Then, the equation for Φ will be