# Ghosts in the self-accelerating DGP branch with Gauss–Bonnet effect

## Abstract

The Dvali–Gabadadze–Porrati brane-world model provides a possible approach to address the late-time cosmic acceleration. However, it has subsequently been pointed out that a ghost instability will arise on the self-accelerating branch. Here, we carefully investigate whether this ghost problem could be possibly cured by introducing the Gauss–Bonnet term in the five-dimensional bulk action, a natural generalization to the Dvali–Gabadadze–Porrati model. Our analysis is carried out for a background where a de Sitter brane is embedded in an anti–de Sitter bulk. Our result shows that the ghost excitations cannot be avoided even in this modified model.

## Keywords

Ghost Ghost Mode Ghost Problem Gaussian Normal Coordinate Ghost Excitation## 1 Introduction

In recent years, the late-time cosmic acceleration has been confirmed by several observational pieces of evidence [1, 2, 3, 4, 5, 6]. This important discovery leads to one of the great puzzles in cosmology, and various plausible models have been developed to unravel the nature of such a late-time speed-up over the last decade. There have been many attempts at building up reasonable and consistent models by modifying the standard cosmology, which can be roughly categorized into two major directions: one is to introduce a dominant dark energy component in the Universe (see, e.g., Ref. [7]), while the other is to modify Einstein’s general relativity at large scales (see, e.g., Refs. [8, 9, 10]).

An intriguing brane-world scenario proposed by Dvali, Gabadadze, and Porrati (DGP) provides a new mechanism with an induced gravity (IG) term, i.e., a four-dimensional (4D) Ricci scalar, included in the brane action [11]. The IG term is expected to arise as a quantum correction due to the matter field on the brane [12], and it makes possible to reproduce the correct 4D Newtonian gravity at short distances even if the bulk is a five-dimensional (5D) Minkowski space-time with an infinite size [11]. The promising feature of the DGP model is that, when generalized to a Friedmann–Lemaître–Robertson–Walker brane with ordinary matter on it, one of its solutions, called the self-accelerating branch, will become asymptotically de Sitter in the far future, giving rise to a late-time accelerating phase without needing to introduce additional substances on the brane that violates the strong energy condition [13, 14].

Despite this advantage, it was pointed out later on that the self-accelerating branch is plagued with a ghost instability [15, 16, 17, 18, 19, 20]. The spin-2 perturbations in this branch, viewed as an effective 4D massive gravity theory on a de Sitter background, are composed of a tower of infinite Kaluza–Klein (KK) massive gravitons. Then the mass of the lowest mode *m* is within the range \(0<m^2<2H^2\) if the brane tension is positive, where *H* is the Hubble parameter, and thus there will be a spin-2 ghost excitation in its helicity-0 component [21].^{1} On the other hand, if the brane tension is negative, the lowest mass is larger than the critical scale, i.e., \(2H^2<m^2\), but the spin-0 perturbation, associated with the brane-bending mode, becomes a ghost instead [17]. In the specific case without brane tension, the lowest mass is equal to the critical scale. Even in this marginal case a detailed analysis shows the existence of a ghost from the mixing between the spin-0 sector and the helicity-0 part of the spin-2 sector [18]. Furthermore, the appearance of ghosts in the DGP self-accelerating branch cannot be eliminated even by invoking a second brane in the bulk with a stabilization mechanism [24]. For more discussions on DGP ghosts, please see Ref. [20] and the references therein. Nonlinear instabilities of the model have also been discussed in Refs. [25, 26, 27].

In this paper, we will investigate the possibility of avoiding the ghost in a generalized DGP model. A natural generalization to the DGP gravitational action, based on its higher-dimensional nature, is by adding the Gauss–Bonnet (GB) term to the original 5D bulk action [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. This modification then yields the most general field equation for the bulk metric with its derivatives only up to the second order [28]. Moreover, this GB term keeps the bulk theory ghost-free and arises as the leading-order correction to the low-energy effective action of the heterotic string theory [29, 30], and furthermore, it plays an essential role in the Chern–Simons gauge theory of gravity [31, 32, 33].

This approach has been first discussed in Ref. [34]. Modes that have the potential to be ghosts are (i) the helicity-0 excitation of the lightest KK graviton and (ii) the brane-bending mode. In Ref. [34], considering a model with a minus sign in front of the bulk Einstein–Hilbert term, the authors succeeded in eliminating these modes. However, considering only an Einstein–Hilbert term in the 5D action even with a minus sign generates ghost modes for the graviton in the bulk. The ghosts can be eliminated by introducing a GB term with a negative prefactor and by taking the anti-de Sitter (AdS\(_5\)) vacuum called the GB vacuum. It is a kind of a ghost condensation of spin-2 particles. Therefore, their model is ghost-free both in the five-dimensional theory and in the four-dimensional effective theory.

The purpose of this paper is, in contrast, confirming if the straightforward extension of DGP model with the GB term still bothers the ghost mode. As discussed in Ref. [34], we expect that the discrete light mode of the KK tower of the graviton and the brane-bending mode to appear in the four-dimensional effective action. Even though these two modes have the potential to be a ghost, their existence does not necessarily imply a ghost excitation. The massive graviton is a ghost excitation only if its mass squared is less than \(2H^2\), while the brane-bending mode becomes a ghost only if the kinetic term has a wrong sign. In the original DGP model, the ghost conditions have been carefully checked [15, 16, 17, 18, 19, 20], and one of the above mentioned condition is always satisfied, i.e. one of the ghost modes appears. This means that the flip of the sign in the kinetic term of the brane-bending mode happens when the squared value of the mass of the lightest KK graviton is \(2H^2\). Nevertheless, its reason is still mysterious [24] and there is no reason why in the extended models the same happens. Moreover, we may expect that the GB terms give a large correction to the self-accelerating branch. With GB corrections, there are three branches of solutions; two branches appearing in the original DGP (the normal branch and the self-accelerating branch) and one additional branch called the GB branch. The transition into the GB branch appears at the high energy region of the self-accelerating branch, and thus it is natural to expect that the self-accelerating branch is largely modified. Therefore, a detailed analysis is needed to confirm the existence of the ghost. Then we study herein the linear perturbations around the background given by (2.8) and (2.9) with \(\epsilon =+1\) which includes as a particular case the self-accelerating branch, and carefully examine whether or not the ghost excitations appearing in the DGP model could be possibly evaded in this framework. We show that even by including the GB term into the bulk, the ghost excitations are still present in this model.

The outline of the paper is as follows. In Sect. 2, we consider the generalized DGP model with the GB term as well as a cosmological constant in the bulk. We then review the background solutions of this system. In Sect. 3, we study the linear perturbations over an AdS\(_5\) bulk with a de Sitter brane within the model introduced in Sect. 2. In Sect. 4, we analyze the effective action for these perturbations, from which we examine the existence of the ghosts in this model. Finally, we present our summary in Sect. 5.

## 2 The model

*M*is split into two regions by a brane hypersurface \(\Sigma \), and the two sides of the brane are denoted by \(\Sigma _{\pm }\). The Latin indices \(a,b,c,\ldots ,\) run from 0 to 4, while the Greek indices \(\mu ,\nu ,\ldots ,\) run from 0 to 3. \(^{(5)\!}g_{ab}\) is the five-dimensional metric, and \(g_{ab}=\,^{(5)\!}g_{ab}-n_an_b\) is the induced metric on the brane, with \(n^a\) being the unit normal vector to the brane; \(\mathcal {R}\),

*R*, \(\kappa _5^2\), \(\Lambda _5\,(<0)\), \(\lambda \), and \(\fancyscript{L}_m\) are the 5D Ricci scalar, the 4D Ricci scalar of the induced metric, the bulk gravitational constant, the bulk cosmological constant, the brane tension, and the matter Lagrangian on the brane, respectively. The GB parameter is denoted by \(\alpha \) (\(\ge \)0), which has the dimension of length square, and the strength of the IG term is characterized by a dimensionless parameter \(\gamma \). Moreover, the second term in Eq. (2.1) corresponds to the generalized York–Gibbons–Hawking surface term [49, 50, 51, 52], where \(K_{\mu \nu }\) is the extrinsic curvature, \(G_{\mu \nu }\) the Einstein tensor of the induced metric, and

*J*the trace of

^{2}\(0<\mu ^2<1/4\alpha \) accordingly. Moreover, this background can be described by the bulk metric

*n*(

*y*) is given by

## 3 Perturbed equations

*y*. However, if we choose the TT gauge for the metric perturbations as in Eq. (3.1), the brane position cannot be fixed at \(y=0\) but will in general reside at \(y=\xi (x^{\mu })\) deviating slightly from the unperturbed position.

*F*:

*m*is the mass eigenvalue. Then the field equation (3.15), in terms of the KK decomposition (3.17), reduces to the simpler form,

*F*. Furthermore, we have also applied the operator decomposition as follows:

## 4 Effective action and existence of ghosts

*s*, respectively. Therefore, the induced metric perturbations \(\bar{h}^b_{\mu \nu }\) can be expressed as

The surprising fact is that the sign flip of the first term in the bracket of Eq. (4.6) always happens with the transition of branches, that is, they are mysteriously related. The lower bound of the inequality (4.7) has been already found in the original DGP model, while the upper bound the inequality (4.7) is shown here for the first time. Then, although the GB branch (with \(\epsilon =+1\)) is a theoretical object and can never describe our Universe, it is interesting that this branch can be ghost-free. Indeed, the GB branch despite having both modes (helicity-0 of the spin-2 sector and the brane-bending mode) can be ghost-free, unlike the self-accelerating branch where either the brane bending or the helicity-0 of the spin-2 is a ghost. The normal branch has a similar behavior to that of the GB branch. In summary, it might be possible to construct a viable cosmological model where despite the existence of the brane bending and the helicity-0 modes, there is no ghost. This may imply the possibility of a ghost-free interesting solution for \(\epsilon =+1\) with a nontrivial modification of the DGP model.

## 5 Summary

In this paper, we looked into a generalized DGP brane-world scenario with a GB term as well as a cosmological constant both incorporated in the bulk action. To check whether this framework can possibly provide a way out of the DGP ghost instability, we have studied the linear perturbations around a de Sitter self-accelerating brane embedded in an AdS\(_5\) bulk space-time. Having the linear perturbations analyzed in this system, we end up with the effective induced metric perturbations on the brane, Eq. (3.23), the physical degrees of freedom in which, as long as none of the KK modes has a critical mass, \(m^2\ne 2H^2\), can be effectively described in terms of the massive KK tower of the spin-2 gravitons as well as the spin-0 excitation associated with the brane-bending mode. Moreover, in contrast with the Fierz–Pauli model for the spin-2 field, gravity in this system can couple to matter with nonzero trace of the energy-momentum tensor when \(m^2=2H^2\), at which the spin-2 and the spin-0 perturbations are degenerate. Therefore, one can no longer divide the degrees of freedom into the spin-2 and the spin-0 sector at this critical scale.

It has been shown that the massive spin-2 field contains a ghost excitation in its helicity-0 component if the mass is in the range \(0<m^2<2H^2\) [21]. Thus, from the massive gravity theory viewpoint, if the mass of the lightest KK mode here is within this forbidden range, there will be a spin-2 ghost excitation present in this system. On the other hand, provided that the mass of the lightest mode becomes larger than the critical scale, \(2H^2<m^2\), although the spin-2 sector becomes healthy in this case, the spin-0 mode is shown to be a ghost instead, similarly to the DGP model. For the specific case where the lightest mass is equal to the critical scale, \(m^2=2H^2\), whether or not a ghost exists in this model cannot be verified rigorously through the method we used here. Presumably, this model still contains a ghost in this marginal case as happens in the DGP model [18]. However, this specific fine-tuning condition, i.e., \(m^2=2H^2\), is easily broken provided that we consider physical matter fields on the brane, in which case the Hubble parameter in general varies with time. As a result, the DGP ghost instability at the level of linear perturbations still cannot be eliminated by invoking the GB term in the bulk action.

Our result shows that in the self-accelerating branch the ghost mode always appears, while in the other branches (with \(\epsilon =+1\)) we have the ghost-free parameter range, although these branches cannot describe our Universe. This is because the sign of the kinetic term of the brane-bending mode not only depends on the mass of the lightest KK graviton but also seems to be related to the branches. We can see it from Eq. (4.6). The sign of the first term in the bracket of Eq. (4.6) is positive for the self accelerating branch and negative for the other two branches. Therefore, the brane-bending mode has both informations of the branch and the value of the lightest KK mass. This relation might be important for the future understanding of the origin of the ghost. Finally, could other generalizations of the DGP model along the line of Refs. [62, 63] appease the ghost problem? We leave this question to a future work.

## Footnotes

## Notes

### Acknowledgments

Y.W.L. is supported by Taiwan National Science Council (TNSC) under Project No. NSC 97-2112-M-002-026-MY3. K.I. is supported by Taiwan National Science Council (TNSC) under Project No. NSC101-2811-M-002-103. M.B.L. is supported by the Basque Foundation for Science IKERBASQUE. She also wishes to acknowledge the hospitality of LeCosPA Center at the National Taiwan University during the completion of part of this work and the support of the Portuguese Agency “Fundação para a Ciência e Tecnologia” through PTDC/FIS/111032/2009. This work was partially supported by the Basque government Grant No. IT592-13. P.C. is supported by Taiwan National Science Council (TNSC) under Project No. NSC 97-2112-M-002-026-MY3, by Taiwan National Center for Theoretical Sciences (NCTS), and by U.S. Department of Energy under Contract No. DE-AC03-76SF00515.

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