DGP Specteroscopy
Abstract:
We systematically explore the spectrum of gravitational perturbations in codimension1 DGP braneworlds, and find a ghost on the selfaccelerating branch of solutions. The ghost appears for any value of the brane tension, although depending on the sign of the tension it is either the helicity0 component of the lightest localized massive tensor of mass for positive tension, the scalar ‘radion’ for negative tension, or their admixture for vanishing tension. Because the ghost is gravitationally coupled to the branelocalized matter, the selfaccelerating solutions are not a reliable benchmark for cosmic acceleration driven by gravity modified in the IR. In contrast, the normal branch of solutions is ghostfree, and so these solutions are perturbatively safe at large distance scales. We further find that when the orbifold symmetry is broken, new tachyonic instabilities, which are much milder than the ghosts, appear on the selfaccelerating branch. Finally, using exact gravitational shock waves we analyze what happens if we relax boundary conditions at infinity. We find that nonnormalizable bulk modes, if interpreted as phenomena, may open the door to new ghostlike excitations.
1 Introduction
Ever since Einstein introduced his famous “biggest blunder”, the cosmological constant has been one of the most frustrating, yet intriguing aspects of General Relativity (GR). Ironically, just as Einstein needed a to make a static universe, if we take his theory of GR as the description of gravity at the largest scales, we now seem to need a to account for the cosmic acceleration observed at redshifts [1, 2, 3]. Unfortunately, manufacturing a sufficiently small, positive cosmological constant from a consistent theory is not entirely straightforward, to say the least. The methods of effective field theory have so far failed to yield a satisfactory microscopic theory of the cosmological constant [4, 5]. Moreover, while the mystery of the cosmological constant is usually posed as a problem for the field theory of matter, one may even wonder if in fact it might really be related to our formulation of gravity and inertia. Our handson experimental knowledge of gravity conforms with GR at distances between mm [6, 7] and, say, MPc. At these large scales we enter the domain of dark matter, a necessary component of the standard cosmological model needed to explain galactic rotation curves, which cannot be accounted for with GR and baryonic matter alone. At the moment, dark matter still needs to be completely explained by particle physics despite a plethora of reasonable candidates. A popular common theme in recent research is that perhaps it is not matter that is needed, but a modification of Newton’s law and/or gravity at large scales. This idea is not new: ever since galactic rotation curves were found to be inconsistent with the luminous matter, such alternatives have been pursued [8].
While it is natural to hope that modifying gravity could be an interesting alternative to dark matter, why might one hope that it could help with the cosmological constant? To illustrate this, we offer the following simple, heuristic argument. It is clear that in the EinsteinHilbert action, the cosmological constant term appears as the Legendre transform of the field variable :
(1) 
From the canonical field theory rules, this means that this term trades the independent field variable for another independent variable . This is exactly the same as in quantum field theory, where one defines the generating function of the theory by shifting the Lagrangian by a ‘coupling’ . This trades the independent variable for another independent variable . After this transformation, the variable is not calculable; it is an external parameter that must be fixed by hand at the end of the calculation, by a choice of boundary conditions. Once is fixed to some value, is calculable in terms of it. The only difference between the usual field theory Legendre transform and the cosmological constant term arises because of gauge symmetries of GR, which render nonpropagating. It is a pure gauge variable that can always be set to a constant number by a change of coordinates. Therefore the Legendre transformation (1) loses information about only one number, which must be fixed externally: namely, by the value of itself. As a result, in GR the cosmological constant is a boundary condition rather than a calculable quantity. One may then hope that by changing gravitational dynamics one could render propagating, so that, in turn, is also rendered dynamical. This could provide us with new avenues for relaxing the value of . Such hopes have been already expressed before on a few occasions [9, 10, 11]. However, analyzing modifications of gravity systematically, to check if they remain compliant with the tests of GR, hasn’t been easy.
On the other hand, in recent years the braneworld paradigm has emerged as a compelling alternative to standard KaluzaKlein (KK) methods of hiding extra dimensions and a new framework for solving the hierarchy problem. In this approach our universe is realized as a slice, or submanifold, of a higher dimensional spacetime. Unlike in KK compactifications, where the extra dimensions are small and compact, in the braneworld approach they can be relatively large [12, 13], or infinite [14, 15]. We do not directly see them since we are confined to our braneworld, rather, their presence is felt via corrections to Newton’s law. Many of the more fascinating phenomenological features of these braneworld scenarios arise in models of warped compactification. In warped compactifications the scale factor of a fourdimensional brane universe actually varies throughout the extra dimensions, providing us with a new way of making a higher dimensional world appear fourdimensional. In general, one can conceal extra dimensions from low energy probes by either 1) making the degrees of freedom which propagate through the extra dimensions very massive so as to cut the corrections to Newton’s law off at long distances, or 2) suppressing the couplings of the higher dimensional modes to ordinary matter so that the gravitational couplings dominate, ensuring that the corrections to Newton’s law are very small at long distances. The latter case is naturally realized in warped models, so that even infinite extra dimensions may be hidden to currently available probes.
Braneworlds provide a natural relativistic framework for exploring means of modifying gravity. It was quickly realized that by using free negative tension branes, one could alter Newton’s constant at large scales [16]. More dramatically, Gregory, Rubakov and Sibiryakov (GRS) [17] noticed that by combining negative tension branes with infinite extra dimensions, it was possible to “openup” extra dimensions at very large scales, making gravity effectively higherdimensional very far away. However, it was soon discovered by the authors that this model contained ghosts [18]. This was unfortunate since the metastable graviton had many desirable gravitational properties, but from a particle physics point of view the existence of a ghost is disastrous. Soon after, a radically new braneworld model was put forward, the DGP (DvaliGabadadzePorrati) model [19], with graviton kinetic terms on the brane as well as in the bulk. The simpler versions of this theory are described by the action
(2)  
In general, there may be additional terms in the bulk. The key new ingredient here is the induced curvature on the brane. It could be generated, as it was claimed initially, by quantum corrections from matter loops on the brane [20], or again in a purely classical picture of a finite width domain wall^{1}^{1}1Harking back to the early manifestations of braneworlds [21]. as corrections to the pure tension DiracNambuGoto brane action [22, 23]. Furthermore, it is also intriguing to note that induced curvature terms appear quite generically in junction conditions of higher codimension branes when considering natural generalizations of Einstein gravity [24] as well as in string theory compactifications [25]. Using holographic renormalization group arguments [26], DGP was shown to be equivalent in the infrared to GRS, however, crucially, it appeared to be ghostfree, corroborating the perturbative analysis of [27]. This made it seem a real candidate for a new gravitational phenomenology at large distances. The induced curvature term yields a particularly interesting new phenomenon. In the case of a brane in Minkowski bulk it allows for a selfaccelerating cosmological solution [28], for which the vacuum brane is de Sitter space, with constant Hubble parameter , even though the brane tension vanishes.
Clearly, the possibility of a fully consistent explanation of large scale acceleration is extremely exciting. It has generated a great deal of activity and investigation into the DGP setup [29] (for a recent review, see [30]), with an astrophysical emphasis on black hole solutions [31], solar system tests [32], shock wave limits [33], and of course, whether DGP can truly explain dark energy [34]. Although many cosmologists have already embraced the DGP model, it has been found to suffer from various problems. There is the issue of strong coupling [35, 36, 37, 38], related to the feature that the graviton interactions go nonlinear at intermediate scales. More importantly, various investigations pointed out that there are ghosts on the selfaccelerating branch [37, 38, 39], however this debate still persists.
Our aim here is to explore this issue in full detail. Since most of the explicit work on DGP has been done for the simplest case of a brane in flat bulk, with the dynamics given by (2), we will work in the same environment, and start with a review of this case. We will next consider the spectrum of small perturbations of the cosmological vacua of DGP, which describe a de Sitter geometry. One of the confusing aspects of the literature on these braneworld perturbations (as opposed to braneworld cosmological perturbations!) is the alternate approaches of direct ‘handson’ calculations, which analyze the curved space wave operator for the gravitational perturbation directly [14, 40, 41] and the “effective action” approach, which was used to particular effect to confirm the ghost [18] of the GRS model via a radion mode analysis [42]. Naturally these approaches should be entirely equivalent, and we will indeed see that. The technical complications in the identification of the spectrum of DGP gravity arise from the mixed boundary conditions for perturbations that may obscure the computation of the norm.
The relevant modes in the spectrum of perturbations for addressing the concerns about stability are the tensors and the scalars. By going to a unitary gauge, we will see that the tensors are generically organized as a gapped continuum of transversetraceless tensor modes, with polarizations per mass level, and an isolated localized normalizable tensor, which lies below the gap. On the normal branch, this localized tensor is massless, implying that it has only two helicity2 polarizations; on the selfaccelerating branch it is massive, with for positive tension, and has polarizations. When the brane tension is positive, the helicity0 mode is the ghost, precisely because its mass sits in the region prohibited by unitarity, explored in [43, 44, 45, 46]. Furthermore, the propagating scalar mode, or the ‘radion’, is tachyonic. This tachyonic instability of scalar perturbations is very generic, and by and large benign (see section 3.2). Moreover, the tachyonic scalar completely decouples on the normal branch in the limiting case where the bulk ends on the horizon^{2}^{2}2We remind the reader that the situation here is similar to the single positive tension brane in the RS model where the radion also decouples.. On the selfaccelerating branch, the scalar mode remains tachyonic but mostly harmless for positive tension branes, but as the tension vanishes it mixes with the helicity0 tensor, and prevents the ghost from decoupling even in the vacuum by breaking the accidental symmetry of the massive tensor theory in de Sitter space in the limit , studied in partially massless theories [43, 45, 46]. This mode becomes a pure and unadulterated ghost when the brane tension is negative, because it contains the brane Goldstone mode which does not decouple in a way similar to the GRS model [17, 18], consistent with the claim of [37]. Thus the selfaccelerating solutions always have a ghost, and therefore do not represent a reliable benchmark for an accelerating universe in their present form. On the other hand the normal solutions are ghostfree, and thus may be useful as a model of gravitational modifications during cosmic acceleration.
Our calculations further allow us to extend the analysis to perturbations which are not symmetric around DGP branes. This symmetry can be relaxed for braneworld models^{3}^{3}3The RandallSundrum model [13, 14] was symmetric by construction, enabling to interpret it as a dual AdS/CFT with a UV cutoff and coupling to gravity [47]., and actually this may be a more natural setting for the DGP setup which is more closely analogous to finite width defects or quantum corrected walls. In fact, in general braneworld models, when the requirement of symmetry is dropped one can get a whole range of interesting gravitational phenomenology, including selfacceleration, without appealing to induced gravity [48, 49]. We will show here that if antisymmetric modes are allowed, then in addition to the ghost, there is an extra excitation which corresponds to the free motion of the DGP brane. This mode is tachyonic, and while it decouples on the normal branch in the single brane limit, it persists on the selfaccelerating branch of DGP solutions. Nevertheless it still remains tame, since the scale of instability is controlled by the Hubble parameter, and so the instability may remain very slow.
The presence of the ghost in the description of the selfaccelerating solutions of DGP indicates that the instability originates from the ‘reduction’ of the theory, and may not really represent a fundamental problem of the bulk setup. A different prescription for boundary conditions might be able to circumvent the contributions from the brane localized ghost. However this requires rather special boundary conditions very far from the brane mass, that would not normally arise dynamically in a local theory. They allow the leakage of energy to, or from, infinity. Worse yet, an explicit exploration of potentials of relativistic sources shows that in this case other modes behave like ghosts, if interpreted in the language. We can see this directly from the gravitational shock wave solutions which include the contributions from the modes that are not localized on the brane.
The paper is organized as follows. In the next section we will review some of the salient features of the DGP model, describing its two branches of background solutions, the normal branch and the selfaccelerating branch. In section 3, we will discuss the perturbation theory around the cosmological vacua of DGP, and identify its occult sector by an explicit calculation. In section 4, using gravitational shock waves, we will consider what happens when we include the contributions from nonnormalizable bulk modes to the long range gravitational potential of brane masses. We will summarize in section 5.
2 What are DGP braneworlds?
We will work with the simplest and most explicit incarnation of DGP, where our universe is a single 3brane embedded in a 5 dimensional bulk spacetime. The bulk is locally Minkowski and the brane carries the curvature of the induced metric as well as the brane localized matter. The induced curvature terms will generically arise from the finite brane width corrections. The brane may be viewed as a function source in the bulk Einstein equations, whose dynamics ensues from the total stressenergy conservation that follows from the covariance of the theory. Alternatively the brane may be treated as a common boundary of two distinct regions, and in the bulk , which are on the different sides of the brane . The boundary conditions at the brane are given by the Israel equations [50], which correspond precisely to the brane equations of motion. These two approaches are physically completely equivalent because the theory is completed with the inclusion of the GibbonsHawking boundary terms [51], which properly covariantize the bulk EinsteinHilbert action in the presence of a boundary. As a result, varying with respect to the metric gives the correct boundary equations as well as the correct bulk. The simpler function form of the field equations then corresponds to the unitary gauge, realized by going to brane Gaussiannormal coordinates, which essentially describe the brane’s rest frame in the bulk, and then gauge fixing residual gauge invariance.
The dynamics of the model can therefore be derived from the action
(3) 
Here is the bulk metric with the corresponding Ricci tensor, (in ). The induced metric on the brane is given by and its Ricci tensor is , while is the brane tension. The extrinsic curvature of the brane is given by , where is the Lie derivative of the induced metric with respect to unit normal, , oriented from into ; is the jump of from to , and is the Lagrangian of brane localized matter fields, with vanishing vacuum expectation value, because the brane vacuum energy was explicitly extracted as tension.
In what follows we will use different gauges for the bulk geometry, because the brane GaussianNormal gauge is very convenient for counting up the modes in the spectrum of the theory, while other gauges may be easier to compute the effective actions for particular modes. Thus, thinking of the solutions geometrically as a bulk in which the brane moves, we will write the field equations which follow from (3) as separate bulk and brane equations of motion respectively. These are valid in an arbitrary gauge, and may be thought of as a breakdown of the full set of field equations on a space with a boundary, where the boundary conditions describe a codimension1 brane. The bulk equations of motion are simply the vacuum Einstein equations,
(4) 
whereas the brane equations of motion are given by the Israel junction conditions [50],
(5) 
where
(6) 
explicitly does not include the brane energymomentum, .
In most of what follows we will impose orbifold symmetry about the brane (section 3.5 will deal with general perturbations). In other words, we will identify and , restricting the dynamics to the symmetric action given by
(7) 
where . The bulk equations of motion (4) are of course unchanged, while the brane equations of motion simplify to
(8) 
2.1 Background solutions
Cosmological DGP vacua describe tensional branes in locally Minkowski patches glued together such that the jump in extrinsic curvature matches the tension and the intrinsic Ricci curvature contributions as in Eq. (8). The solutions can be easily constructed by taking a bulk geometry which solves the sourceless bulk equations (4), and slicing it along a trajectory which solves (5). Then becomes the cosmological scale factor and the comoving time coordinate. Such techniques have been used before in the RS2 framework [52, 53]. When the brane only carries nonzero tension, its worldvolume is precisely a de Sitter hyperboloid representing the de Sitter embedding in a Minkowski space as required by the symmetries of the problem [54]. This solution generalizes the geometry of VilenkinIpserSikivie inflating domain walls in [55], and was in fact also found in [56] in the context of finite thickness domain walls.
In conformal coordinates , the full background metric is given by
(9) 
where
(10) 
and
(11) 
The bulk spacetime, , is the image of the line , with the brane positioned at . In DGP brane induced gravity theory there exist two distinct branches of bulk solutions, labelled by . The solution with is commonly referred to as the normal branch whereas the solution with is referred to as the selfaccelerating branch, a terminology which will become transparent shortly. The brane metric in (10) represents the de Sitter geometry in spatially flat coordinates, which cover only one half of the de Sitter hyperboloid. The complete cover with global coordinates involves the metric describing a sequence of spatial spheres , of radius and spatial line element , which initially shrink from infinite radius to radius , and then reexpand back to infinity.
The intrinsic curvature is given by the tension, as dictated by the brane equations of motion (a.k.a. brane junction equations) (5) at ,
(12) 
The solutions are
(13) 
This equation suggests that there are in fact four possible values of the intrinsic curvature. However this is not the case. It is easy to see that only two of these solutions are independent. Indeed, note that a bulk reflection and a time reversal map two of the solutions (9,10) with onto the solutions with simultaneously reversing the sign of . Thus without any loss of generality we fix the signs by requiring that a positive tension corresponds to positive intrinsic curvature , so that
(14) 
We can embed these solutions in the bulk as in figure 1 [33]. For , or “selfaccelerating” branch, we retain the exterior of the hyperboloid. For , or the “normal branch”, we keep the interior of the hyperboloid. It is now clear whence the terminology: on the selfaccelerating branch, even when the tension vanishes, , the geometry describes an accelerating universe, with a nonvanishing curvature produced solely by the modification of gravity. The scale of the curvature needs to be specially tuned to the present horizon scale of eV, which corresponds to about MeV [28, 29], but once this is done one may hope to explain the current bout of cosmic acceleration even without any Standard Model vacuum energy. The selfaccelerating branch of solutions are a distinct new feature of the DGP model, they do not exist on symmetric brane without the induced gravity terms on the brane [54, 28]. However, related solutions may arise in theories with asymmetric bulk truncations [48, 49].
2.2 How do we obtain gravity in the DGP model?
A crucial question is: given the cosmological DGP vacua reviewed above, how could there be a low energy gravitational force between masses inhabiting them? Unlike in RS2, for solutions given by (10) and (12), the ‘apparent’ warping of the bulk cannot play a significant role in manufacturing gravity at large distances. In RS2 bulk gravitational effects pull the KK gravitons away from the brane, strongly suppressing their couplings to brane localized matter. As a result, the extra dimension is hidden. That does not happen here because the bulk in (10) is locally flat. Moreover, on the selfaccelerating backgrounds the bulk volume is infinite, and so the graviton zero mode is decoupled: it is not perturbatively normalizable, and the mass scale which governs its coupling diverges. Although the bulk volume for the normal branch solutions is finite for finite , and there is a normalizable graviton mode, its coupling^{4}^{4}4This formula is precisely the analogue of the Gauss law relation between bulk and effective Newton’s constant in models with large extra dimensions [12]. is , and so it also decouples in the limit [33]. In fact, from the general embedding of a de Sitter hyperboloid in Minkowski space (10) we see that the limit corresponds to taking the radius of extrinsic curvature of the hyperboloid on the normal branch to infinity, de facto pushing it to the spatial infinity of Minkowski space. In this limit the bulk volume between the brane and the horizon diverges, which is why the zero mode graviton decouples. This agrees with the perturbative analysis of the case of [19, 27] where the zero mode graviton was completely absent. Hence gravity ought to emerge from the exchange of bulk graviton modes.
Suppose first that the graviton kinetic terms reside only in the bulk. In an infinite bulk, a typical bulk graviton sourced by a mass on the brane will not venture too far from the brane because it would cost it too much energy. Nonetheless if kinetic terms reside only in the bulk, a typical bulk graviton would still peel away from the brane and explore the region of the bulk around the brane out to distances comparable to the distance between the source and a probe on the brane. The momentum transfer by each such virtual graviton to the brane probe would be , where is the momentum along the brane, as dictated by the graviton propagator and brane couplings. Thus the gravitational potential would scale as , and the resulting force as . Such forcedistance dependence would reveal the presence of the extra dimension. This would remain true even when on the normal branch. Although a zero mode is present in this case, it cannot conceal the extra dimension because it would still be too weakly coupled to provide the dominant contribution to the long range force at subhorizon scales.
The induced curvature terms on the brane change this in DGP. In order for this trick to work, one needs to be big. In this case, the brane localized kinetic terms effectively pull the zero mode gravitons closer to the brane, making their exploration of the bulk at distances shorter than energetically costly [19, 27]. This alters the scaling of the momentum transfer to for momenta , which in turn produces a force which scales as . This is manifest from the explicit form of the graviton propagator projected on to the Minkowski brane (i.e. the limit of the normal branch solutions of (10) and (12)) [19, 27]:
(15) 
From the point of view, the graviton resonance which is exchanged is composed of massive tensor modes, and so it will contain admixtures of longitudinal gravitons. This is encoded in the propagator (15) in the coefficient of the last term of the spin projector, as opposed to which appears in the linearized limit of standard GR. This difference is an example of the venerated Iwasakivan DamVeltmanZakharov (IvDVZ) discontinuity of modified gravity [57], and signifies the persistence of a scalar component of gravity in the theory, that could conflict with the known tests of GR. However, it has been argued for massive gravity [58] and similarly for the DGP model [59] that the extra scalar may be tamed by nonlinearities once the correct background field of the source is included. The idea is that the perturbative treatment of the scalar graviton breaks down at a distance scale first elucidated by Vainshtein [58]. For DGP, for a source of mass , this new scale is given by [60, 61]. Below that distance, one can’t trust linearized perturbation theory and must resum the background corrections, which should presumably decouple the scalar graviton mode. Similar weakening of the scalar graviton coupling may occur at cosmological scales if the universe is curved.
This scale dependence of the scalar graviton couplings has very interesting and important implications for the DGP setup. It has been pointed out [35, 36, 37, 38] that the effective field theory description of DGP gravity will suffer from a loss of predictivity due to the problems with strong couplings at distances , which could be much larger than the naive UV cutoff. The most recent analysis of this issue [38] however suggests that the brane nonlinearities may push the scale of strong coupling down, to about on the surface of the Earth, possibly making DGP marginally consistent with current table top experimental bounds [6, 7]. In what follows we will assume this claim [38] and imagine that we work in the perturbative regime of DGP, although we feel that this issue deserves further attention. We note that the exploration of DGP with gravitational shock waves [33] shows that the scalar graviton decouples from the background of relativistic sources, indicating that the coupling is effectively suppressed by the ratio of of the source. Note, that this is not enough to ascertain that a theory is phenomenologically safe. For example, a BransDicke theory will admit identical shock waves as ordinary GR for any value of the BransDicke parameter , while the observations require that . Thus one still needs to study the model for slowly moving sources to check if the predictions agree with observations. However one may hope that the strong coupling problems might be resolved in a satisfactory fashion. After all, the shock waves [33] remain valid down to arbitrarily short distances from the source, behaving much better than they are entitled to given the concern about the strong coupling problems.
In what follows we will focus on uncovering the ghosts (and/or other instabilities) on the selfaccelerating branch, and a discussion of their implications for DGP. Before we turn to this, we should stress that there is no technical inconsistency between our results and the earlier claims that there are ghostfree regimes in DGP [19, 27]. Indeed: starting with the backgrounds of the family (10,12) and fixing and , the only way to consistently take the limit to is to pick the normal branch solutions and dial the brane tension to zero. In this way one reproduces the brane backgrounds with fixed , that were studied in [19, 27]. Moreover, ghosts may also be absent if the brane geometry is anti de Sitter, as opposed to dS [62]. Thus the results of the perturbative analysis of [19, 27], implying the absence of ghosts and other instabilities on branes, applies only to the normal branch backgrounds of DGP (10), (12). In fact, our results will confirm this for the general backgrounds of the normal branch, showing that they are ghostfree. However the analysis of [19, 27] has nothing to say about the selfaccelerating branch solutions, and specifically about the limit, that describes a universe where cosmic acceleration arises from modification of gravity alone. In what follows we will confirm that in all those cases there are ghosts, which invalidate the selfaccelerating branch solutions in their present form as realistic cosmological vacua.
3 The occult sector of DGP
We now turn to the exploration of the spectrum of light modes in the gravitational sector of DGP, around the cosmological vacua (10), (12). We will confirm that there are ghosts in the effective field theory description on the selfaccelerating branch of DGP solutions. More specifically: in the effective field theory which describes the perturbative regime of selfaccelerating branch of DGP backgrounds (10), (12) between the Vainshtein scale and the scale of modification of gravity there are scalar fields with negative, or vanishing, kinetic term around the vacuum, which couple to the branelocalized matter with at least gravitational strength. Now, this may appear surprising at the first glance: there are no ghosts in the action (3) of the full bulk theory. Indeed, the full bulk Lagrangian in (3) does not appear to contain any instabilities. However, the background solutions (10), (12) of (3) involve an endoftheworld brane, which is a dynamical object, whose worldvolume is determined by (5). The problems arise because the brane will curl up and wiggle when burdened with a localized mass, in a way that alters the gravitational fields of the source mass, spoiling the guise of the theory. Thus the perturbative ghost encountered in theory is really a diagnostic of the failure of the perturbation theory to describe the dynamics of the long range gravity on the selfaccelerating solutions. Thus although the applications [28, 29, 63] of the selfaccelerating solutions to cosmology are interesting and tempting, the presence of the ghost renders them unreliable at the present stage of understanding of the theory, and hence de facto inadequate as a method of accommodating the present stage of cosmic acceleration.
In the following subsections we will identify the independent degrees of freedom describing small perturbations around DGP vacua in both branches, derive their linearized equations of motion and solve them. We will then compute the four dimensional effective action, isolate the ghost of the theory, and discuss its consequences.
The physical interpretation of these solutions is based on the mathematical analysis of a differential operator derived by considering perturbations of Einstein’s equations: the Lichnerowicz operator . This operator acts on a fivedimensional spacetime with a timelike boundary (the brane). We can solve these perturbation equations in whatever gauge we like, however, in order to get a braneworld interpretation of the results, the cleanest procedure we can follow is to separate this problem (operator plus space on which it acts) into a direct sum of a purely fourdimensional operator acting on a fourdimensional spacetime, and a selfadjoint ordinary differential operator acting on the semiinfinite real line. Obviously this latter operator acts on the space perpendicular to the brane, and hence to really benefit from this factorization, in these coordinates the brane should be held at a fixed coordinate position. Once we have made this decomposition, we will be able to identify the physical states and their norms from the braneworld point of view. To this end, we should write the perturbation in its irreducible components with respect to the braneworld, correctly identify the degrees of freedom corresponding to “motion” of the brane, and reduce our perturbation equations to a selfadjoint form.
3.1 Learning to count: mode expansion
We turn to the linearized perturbations about the background metric (9), (10), (12), defined by the general formula
(16) 
where we use the shorthand . Note that as specified in (9), (10), (12). From now on, we will raise and lower indices () with respect to the de Sitter metric , and designate de Sitter covariant derivatives by . Our normalization convention for the perturbations in (16) reflects afterthefact wisdom, in that it simplifies the bulk mode equations to a Schrödinger form, as we will see later on.
Since the spacetime ends on the brane, if we fix the gauge in the unperturbed solution (9), (10), (12) such that the brane resides at , a general perturbation of the system will also allow the brane itself to flutter, moving to
(17) 
The explicit expressions for the perturbations are obviously gaugedependent. Now, to consider their transformation properties under diffeomorphisms
(18) 
we should first classify them according to different representations of the diffeomorphism group as
(19) 
comprising in total tensor + vector + scalar towers towers of degrees of freedom plus one more scalar, i.e. precisely the number of independent fluctuations of a symmetric bulk metric and the brane location. Clearly, not all of these degrees of freedom are physical: some can be undone by diffeomorphisms (18). Indeed, we can easily derive the explicit infinitesimal transformation rules,
(20) 
where we have used that .
In order to have a clear braneworld interpretation of variables, we find it convenient to work in the Gaussiannormal (GN) gauge (see e.g. [64]), in which any orthogonal component of the metric perturbation vanishes. Given any perturbation (19), we can transform to a GN gauge by picking the gauge parameters
(21) 
which set and to zero. This still leaves us with components of and the brane location (omitting the primes), accompanied by residual gauge transformations
(22) 
which can remove several more mass multiplets from the perturbations. However these could only be zero modes of some of the bulk fields, because of the restricted nature of the bulk variation of (22). Rather than completely gauge fix the perturbations now, it is more useful to resort to dynamics to find out which of the modes , are propagating and which are merely Lagrange multipliers. To this end we can first decompose the tensor in terms of irreducible representations of the diffeomorphism group. This yields (for a proof, see Appendix (6))
(23) 
where is a transverse traceless tensor, components, is a Lorentzgauge vector, , with components, and and are two scalar fields (such that they correctly add up to the total of 10 degrees of freedom). , with
To get some feel of the dynamics before looking directly at the field equations, we can check how these modes transform under the residual gauge transformations (22). Substituting the residual gauge transformation (22) into (20), we find that the surviving, symmetric, tensor mode in the GN gauge and the brane location field transform as
(24) 
where , and we have used that . If we further split up the gauge transformation parameter , where , and apply the decomposition (19) of into the irreducible representations and to (24), after a straightforward computation we find that the irreducible representations transform according to
(25) 
Note that while the decomposition (23) of a general into irreducible representations of the diffeomorphism group is kinematically unique, implying the breakdown of the residual gauge transformations as in (25), it does not  in general  guarantee that different modes won’t mix dynamically. Indeed, in writing (23) we are implicitly assuming that different irreducible transformations live on different mass shells, and hence cannot mix dynamically at the quadratic level. This can be glimpsed at, for example, by noting that while the symmetries of the problem allow us to write the couplings like etc, the TT conditions for would imply that such couplings are pure boundary terms for nonsingular couplings . While this is true in general, the situation is considerably subtler when the representations become degenerate. In this instance the decomposition (23) requires more care. New accidental symmetries mixing different representations, notably tensor and scalar, may arise, modifying (25) and dynamically mixing the modes. This occurs in the vanishing brane tension limit on the selfaccelerating branch of DGP. We will revisit this limit in more detail later on.
Keeping with the general situation for now, the transversetraceless tensor is gauge invariant, while the vector and the scalars are gauge dependent – we can gauge away the zero mode of the vector and one of the scalars. In addition, we see how the motion of the brane can be gauged away, choosing to set the location of the brane to . By doing this, we are explicitly choosing coordinates which are branebased, and the metric perturbation (23) has an explicit term describing the brane fluctuation. In the branebased approach, we have completely and rigorously separated the Lichnerowicz operator into brane parallel and transverse parts. However: once we have taken these coordinates, we do not have the liberty of making residual gauge transformations parameterized by in (25), because they would move the brane from the coordinate origin. In effect, the brane is tied to the dynamical fields and in the bulk, but its fluctuation turns into a Goldstone boson of the system. We emphasize that this is a gauge choice. We are choosing the braneGN gauge to make the separation of the Lichnerowicz operator mathematically clean. However, one can also choose to allow the brane to fluctuate freely (and indeed the effective action computation is better done this way) by having a bulkGN gauge, with the gauge freedom in (25), and the brane sitting at . In this case, there are no fixed terms in the perturbation, and the brane motion enters into the boundary condition via evaluation of the background solution at . The actual equations of motion and boundary terms in both gauges are identical, giving the same physical results and the same dynamical scalar fields. Thus, explicitly in braneGN gauge:
(26) 
To proceed with setting up the problem, we derive the field equations for the irreducible modes. Having restricted to the family of braneGN gauge perturbations (26), we can substitute them in the field equations (4), (5) and after straightforward algebra write the linearized field equations in the bulk,
(27) 
where
(28)  
(29)  
(30) 
and
(31)  
and on the brane,
(32) 
Before we proceed with the details of the mode decomposition of this system by direct substitution of (23), we note that the Lorentzgauge vector turns out to be a free field in the linearized theory in flat bulk. Thus the solutions for decouple from the branelocalized matter in the leading order. They are irrelevant for the stability analysis which is our purpose here. Hence we will set from now on, assuming we have arranged for bulk boundary conditions which guarantee this in the linear order.
3.2 Fluctuations around the vacuum
First note that independently of matter on the brane, the and equations must be identically satisfied. In conjunction with the trace of the equation, this can be seen to imply that a gauge can be chosen in which , and . If in addition we have no matter on the brane, then we see that
(33) 
and so the metric perturbation (26) is completely transverse and traceless.
The remaining bulk equations then simplify considerably to give
(34) 
with the boundary condition
(35) 
Now, to solve this equation we should carefully decompose the tensor into orthogonal modes, which in general do not mix at the linearized level. Those are exactly the irreducible representations we discussed previously. Thus using linear superposition, we can expand the general metric fluctuation in and , the latter of which couples to the field through the boundary condition (35), leaving the TTtensors with entirely homogeneous boundary conditions. We therefore write
(36) 
where we have performed the mode expansion , in terms of the modes which satisfy . We have also defined the scalar mode , and separated variables in the scalar field by setting , where is a general tachyonic field obeying
(37) 
This tachyonic mode is present whenever we compactify the theory on an interval with de Sitter boundary branes. It can be traced back to the repulsive nature of inflating domain walls [55]. Here, it is simply an indication that a multide Sitter brane configuration requires a special stabilizing potential, as is familiar already in the context of RS2 braneworld models [65, 66, 67]. This kind of an instability is generically much slower and hence less dangerous than the ghost, as it is governed by the scale that is as long as the age of the universe. When tension is positive, this mode therefore remains largely harmless for the phenomenological applications of the theory. However, on the selfaccelerating branch it mixes with the ghost in the vanishing tension limit on the selfaccelerating branch, and becomes the ghost itself for negative tension, as we will see later.
We now turn to the analysis of the perturbations which form the main part of the propagator, and determine the norm on the transverse space. The bulk field equation (34) and the boundary condition (35) reduce to the boundary value problem
(38) 
which is selfadjoint with respect to the inner product
(39) 
The eigenmodes with different eigenvalues are orthogonal. We choose the normalization such that the discrete modes, if any, satisfy , while the continuum modes satisfy . This is simply a reflection of the fact that far from the brane the bulk modes behave just like bulk plane waves, and the mass is precisely the component of the momentum.
To determine the spectrum of the boundary value problem (3.2), (39) we rewrite the boundary value problem (3.2) as a Schrödinger equation
(40) 
It is now clear that the solutions of (3.2) must fall into two categories: one discrete mode for each branch, localized to the function potential, if it is normalizable according to (39), and a continuum of ‘free’ modes, gapped by .

: the normalizable solution of (3.2) in the bulk, representing a single, light, localized graviton on each branch, with a mass
(41) fixed by the boundary conditions (3.2), and wave function
(42) where and . On the normal branch ,
(43) On the self accelerating branch ,
(44) Herein is our first glimpse of the tensor ghost: for positive tension, the localized light graviton mode on the self accelerating branch lies in the forbidden mass range discussed in [44, 45, 46]. Its helicity0 component is the ghost, as we will review later on (see Appendix (7)).

: the function normalizable modes are
(45) where and . The integration constant which solves the boundary condition (3.2) is
(46)
Turning now to the scalar component , it is not difficult to see that it obeys
(47) 
(equivalent to a mass ). The bulk equation (34) then yields the wave equation for ,
(48) 
The boundary condition (35) enforces a relation between and :
(49) 
The wave function solutions for either of the DGP branches are
(50) 
From (39), the norm is determined by , where the lower limit of integration accounts for the unperturbed location of the brane at , around which we impose the symmetry. Thus the mode is normalizable but the mode is not. We therefore set . This choice, at least in principle, corresponds to prescribing boundary conditions at infinity, which ensure the brane is an isolated system. Thus setting , and separating the variables by setting in (49) we find
(51) 
However, we must be mindful of this choice because of the possible interplay with the brane bending term (25), as we will see next.
Now: on the normal branch , it follows from (25) that the normalizable mode is gaugedependent: in fact, it is of the same form as the branebending mode since it is proportional to . On the other hand the nonnormalizable mode is gaugeinvariant by itself, and so setting it equal to zero is straightforward. Then (51) gives , which means that the brane boundary condition (35) in fact precisely sets the normalizable gaugeinvariant mode to zero. Hence
(52) 
Thus the net effect of the mode is to undo the brane bending. This is because the translational invariance of the branebulk system, which yields the residual gauge symmetry (25) is linearly realized in the presence of the brane, which imposes gaugeinvariant boundary condition, so that the normalizable bulk mode and the brane bending completely compensate each other. Put another way, the only consistent matterfree solution for the normal branch DGP brane is where the brane does not move from , and only TT GN perturbations in the metric are allowed. This, of course, should have been expected all along, as it is just the statement that the radion field decouples in the case of single UV brane with Minkowski or de Sitter geometry embedded in the standard way in Minkowski or space. Here we see explicitly how gauge invariance and normalizability enter this subtle conspiracy to remove this mode, essentially allowing that any scalar bulk perturbation localized to the brane can be bent away.
On the selfaccelerating branch , the situation is very different: now, the normalizable scalar mode is gaugeinvariant by itself. The nonnormalizable mode is not, and so imposing boundary conditions which require breaks the residual gauge invariance (25). The brane bending mode is the Goldstone field of the broken symmetry, and the brane boundary condition (35) for a generic value of (i.e. for nonzero tension) yields
(53) 
which pins the Goldstone to the normalizable gaugeinvariant scalar perturbation :
(54) 
This perturbation represents a genuine radion, or physical motion of the brane with respect to infinity. Although our choice of braneGN gauge fixes the brane to the coordinate position , it does so at the cost of, this time, breaking the residual gauge symmetry generated by in (25) and introducing the explicit “bookkeeping” term in , which is the remnant of the translational zero mode of the brane. Had we instead allowed the brane position to be arbitrary, at , (without the term in (26)), the boundary conditions at would still have had the same form, since the terms would have entered when evaluating the background at nonzero . Both approaches are completely equivalent, the former being more suitable to a brane based observer and the latter to an asymptotic observer. The gauge transformation between these is a translation, which therefore corresponds to real motion of the brane, just as in the 2brane RS case [41]. The absence of this mode on the normal branch reflects the fact that there is no distinguishable motion of an individual symmetric brane.
When the tension is different from zero, the solutions are precisely the TTtensors of (19) from the previous subsection. The scalar mode has eigenvalue , as seen from (47), and the eigenvalues of the eigenmodes are all different from when . Thus the scalar mode , disguised as the tensor , is orthogonal to all . Hence coincide with the tensors , and so when there is no matter on the brane, the solutions are given by
(55)  
This solution clearly remains valid on the normal branch even in the limit of vanishing tension, , and for the full range of , because when the potentially dangerous terms vanish identically.
However on the selfaccelerating branch where the solution (55) – as it stands – fails when the tension vanishes, , because of the pole in , or , (53), (54). Indeed, (14) implies that when , , and so the parameter in (53) diverges. Thus the mode as given by (54) is illdefined in this limit. At a glance, noting that the coefficient of in (49) vanishes, one may interpret equation (53) as implying , thus fixing the brane rigidly at , and allowing to fluctuate independently of . However, in light or the residual gauge transformations (25) and our gaugefixing , that removed the nonnormalizable gaugedependent bulk scalar, setting also would completely break the residual gauge symmetry group. This is dangerous, since it may miss physical degrees of freedom, which warns us against such a quick conclusion. To see what is really going on we must tread carefully.
What’s going on when the tension vanishes is that the mass of the localized tensor mode on the selfaccelerating branch approaches , as is clear from (41). Further, the bulk wave function of the lightest localized tensor (42) converges to , i.e. it becomes identical to the bulk wave function of the gaugeinvariant scalar mode . Thus the lightest tensor, , and the scalar become dynamically degenerate, and can mix^{5}^{5}5This mixing has been noticed as the resonance instability in the shock wave analysis of [33], and discussed at length in [68]. together: they both solve the field equations and have formally the same tensor structure. Now, it has been noted by Deser and Nepomechie [43] that in the special case when the mass of the massive PauliFierz theory in de Sitter space equals , the theory develops an accidental symmetry[43, 44, 45, 46]. The tensor dynamics becomes invariant under the transformation , where is any solution of the equation , affecting only the helicity0 component of .
Lifting this symmetry to the present case is considerably more intricate because of the degenerate scalar . Noting first that the wave profile of the lightest localized tensor is now , the accidental symmetry of [43] shifts the bulk TTtensor by
(56) 
However given the higherdimensional origin of the perturbations we cannot arbitrarily shift these modes around. The only gauge generators available to us, that could in principle generate such shifts, are the residual gauge transformation rules of (24). However as is clear from (24), none of the residual gauge transformations have the correct bulk wave profile to yield (56). Thus the transformation (56) must be understood as the Stückelberg symmetry of the problem: shifting