PHYSICAL REVIEW D 72, 123511 (2005)

Ghosts in the self-accelerating brane universe

Kazuya Koyama1,2

1Department of Physics, University of Tokyo 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan2Institute of Cosmology and Gravitation, Portsmouth University, Portsmouth, PO1 2EG, United Kingdom

(Received 5 April 2005; revised manuscript received 10 October 2005; published 16 December 2005)

1550-7998=20

We study the spectrum of gravitational perturbations about a vacuum de Sitter brane with the induced4D Einstein-Hilbert term, in a 5D Minkowski spacetime (DGP model). We consider solutions that includea self-accelerating universe, where the accelerating expansion of the universe is realized withoutintroducing a cosmological constant on the brane. The mass of the discrete mode for the spin-2 gravitonis calculated for variousHrc, whereH is the Hubble parameter and rc is the crossover scale determined bythe ratio between the 5D Newton constant and the 4D Newton constant. We show that, if we introduce apositive cosmological constant on the brane (Hrc > 1), the spin-2 graviton has mass in the range 0<m2 < 2H2 and there is a normalizable brane fluctuation mode with mass m2 � 2H2. Although the branefluctuation mode is healthy, the spin-2 graviton has a helicity-0 excitation that is a ghost. If we allow anegative cosmological constant on the brane, the brane fluctuation mode becomes a ghost for 1=2<Hrc < 1. This confirms the results obtained by the boundary effective action that there exists a scalarghost mode for Hrc > 1=2. In a self-accelerating universe Hrc � 1, the spin-2 graviton has mass m2 �2H2, which coincides with the mass of the brane fluctuation mode. Then there arises a mixing between thebrane fluctuation mode and the spin-2 graviton. We argue that this mixing presumably gives a ghost in theself-accelerating universe by continuity across Hrc � 1, although a careful calculation of the effectiveaction is required to verify this rigorously.

DOI: 10.1103/PhysRevD.72.123511 PACS numbers: 98.80.Cq, 04.50.+h, 11.25.Wx

I. INTRODUCTION

The cosmological constant problem is one of the mostdifficult and important problems in particle physics andcosmology [1]. The old problem is the smallness of thecosmological constant compared with the fundamentalscales of particle physics. A natural solution for this prob-lem has been thought to be an exact cancellation of thecosmological constant. However, recent discoveries of theaccelerated expansion of the universe make the problemmore complicated [2]. Within the 4D Einstein theory ofgravity, we do need a tiny cosmological constant in order toexplain the acceleration of the universe. Then we mustintroduce a tiny cosmological constant by hand and accepta fine-tuning [3].

There have been many attempts to modify the 4DEinstein theory of gravity to explain the acceleration ofthe universe instead of introducing the cosmological con-stant [4]. One such attempt was made in the context of thebrane world models, where our universe is realized as a 4Dhypersurface (brane) in a higher dimensional spacetime(bulk). Dvali, Gabadadze, and Porrati (DGP) proposed amodel where the 4D Einstein-Hilbert term is assumed to beinduced on the brane [5]. In this model, the acceleratedexpansion of the universe can be realized by a modificationof Einstein gravity on large scales and we do not need acosmological constant (self-accelerating universe) [6].

Recently several authors claimed that there exist ghost-like excitations in the self-accelerating universe [7,8]. Ifthis is true, it becomes difficult to consider the DGP modelas a consistent model to explain the acceleration of theuniverse. In this paper, we reexamine the existence of the

05=72(12)=123511(5)$23.00 123511

ghost in the DGP model based on a direct claculation of thespectrum of the perturbations.

II. BACKGROUND SPACETIME ANDPERTRUBATIONS

The 5D action describing the DGP model is given by

S �1

2�2

Zd5x

��������gp

R�1

2�24

Zd4x

����������p �4�R

� �Zd4x

����������p

�1

�2

Zd4x

����������p

K; (1)

where � is the tension of the brane, K�� is the extrinsiccurvature and K � K�

�. In this letter, we only consider avacuum energy contribution from matter fields on thebrane for simplicity. We also assume reflection symmetryacross the brane. Then the junction condition must beimposed at the brane as

K�� �

�2

2

���3��� �

1

�24

~G��

�; (2)

where �4�G�� is the Einstein tensor on the brane and

�4� ~G�� �

�4�G�� � �1=3��4�G���. The 4D Einstein tensor

comes from the induced Einstein-Hilbert term. TheFriedmann equation on the brane is given by

�H � rcH2 �

�2

6�; rc �

�2

2�24

: (3)

The 5D solution for the metric with the 4D de Sitter branecan be obtained as [6]

-1 © 2005 The American Physical Society

KAZUYA KOYAMA PHYSICAL REVIEW D 72, 123511 (2005)

ds2 � dy2 � N�y�2���dx�dx�; N�y� � 1�Hy;

(4)

where ��� is the metric for the de Sitter spacetime and thebrane is located at y � 0. If we take the � branch solu-tions, there is a solution for the de Sitter spacetime without�,

H �1

rc: (5)

We call this solution the self-accelerating universe.Let us investigate the perturbations N�y�2��� � h��

about the background de Sitter spacetime. In the following,we assume Hrc � 1 and treat the case Hrc � 1 separately.In addition to the gravitational perturbations h��, we musttake into account a perturbation of the position of the braney � ’�x� [9]. Using the transverse-traceless gauger�h�� � h � 0, the perturbed junction condition is givenby

k�� �Hh�� � rc

�X���h� � �2

4

�T�� �

1

3���T

��

� ��1� 2H rc��r�r� �H2����’;(6)

where k�� � �1=2�@yh��, H � @yN=N on the brane andX�� is given by

X�� � ��4�G�� � 3H2h��

� �1

2��4h�� �r�r�h�� �r�r�h�� �r�r�h�

�1

2����r�r�h�� ��4h� �H2

�h�� �

1

2���h

�:

(7)

The equation of motion for’ is obtained from the tracelesscondition h � 0;

�1� 2H rc���4 � 4H2�’ ��2T

6: (8)

Let us find solutions for the vacuum brane T�� � 0.Using the separation of variables h�� �Rdme���x�Fm�y�, the equation of motion in the bulk is

written as

F00m �1

N2 �m2 � 2H2�Fm � 0; (9)

where prime denotes a derivative with respect to y. Thereare two types of solutions. One type of solution is aninhomogeneous solution sourced by the scalar mode ’.We call this solution the spin-0 perturbation [10]. The othersolution is a homogeneous solution with ’ � 0, which iscalled the spin-2 perturbation. The spin-2 perturbations�� satisfy the junction condition without ’

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0�� � 2H�� � �m2rc��: (10)

We find a tower of continuous Kaluza-Klein (KK) modesstarting from m2 � �9=4�H2 as well as a normalizablediscrete mode m2

d � 0 in the � branch and with

m2d

H2�

1

�Hrc�2 �3Hrc � 1�; (11)

in the� branch forHrc > 2=3 [11]. ForHrc > 1, the massis in the range 0<m2

d � 2H2 where m2d � 2H2 for the

self-accelerating universeHrc � 1 andm2d ! 0 forHrc !

1 [12].In the � branch, there are no normalizable solutions for

the spin-0 perturbations. In the� branch, there is a normal-izable solution given by

h�� �1� 2HrcH�1�Hrc�

�r�r� �H2����’: (12)

This is a solution with m2 � 2H2.

III. BOUNDARY EFFECTIVE ACTION

We can construct the 2nd order action for h�� and ’from the 5D action by extending the result of Ref. [13]. Theresult is given by

�2S � �1

4�2

Zd5x

��������gp

N�4h����5�G��

�1

�2

Zd4x

����������p

LB; (13)

where ��5�G�� is the 5D perturbed Einstein tensor and

LB � k��h�� � kh�1

2H �h2 � h��h���

� �1� 2H rc��h��r�r�’� hrr’� 3H2h’�

� 3H���1� 2H rc�’��4 � 4H2�’�

�2

3T’

�

�1

2�2h��T�� �

rc2h��X���h�: (14)

This action gives the correct equation of motion and thejunction condition for h�� and the equation of motion for’.

We can derive an effective action for the brane fluctua-tion ’ by substituting the 5D solution for h�� given by ’(12) into the 5D action and get the off-shell action for ’ byintegrating out only with respect to the extra coordinate y[14]. This yields the action for ’ in the � branch as

S’ �3H

2�2

�1� 2Hrc1�Hrc

�Zd4x

����������p

’��4 � 4H2�’: (15)

We find that, for Hrc > 1, the kinetic term is alwayspositive, so the brane fluctuation mode ’ is not a ghost.

However, there is a problem for the spin-2 perturbations.We consider only the � branch solutions that include the

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GHOSTS IN THE SELF-ACCELERATING BRANE UNIVERSE PHYSICAL REVIEW D 72, 123511 (2005)

self-accelerating universe. The 4D effective action for thespin-2 perturbations is also obtained in a similar way. Forthe discrete mode with m2

d, we get

S�rc�3Hrc�1�

4�2�3Hrc�2�

Zd4x

����������p

����4�2H2�m2d���;

(16)

where transverse-traceless gauge fixing conditionsr��� � �� � 0 are imposed. This is exactly the sameaction for the spin-2 perturbations in the 4D massivegravity theory where the Pauli-Fierz (PF) mass term isadded to the Einstein-Hilbert action by hand [15]

SM � �M2

8�24

Zd4x

����������p

�h��h�� � h2�: (17)

Note that in the 4D PF theory, the mass term breaks thegauge symmetry for the perturbation and the transverse-traceless conditions are imposed by the equation of motionas constraints. In the limitHrc ! 1, the action approachesthe one for massless spin-2 perturbations. However, thereis a discontinuity between the massless perturbations andthe massive perturbations that is known as the van Dam-Veltman-Zakharov discontinuity [16]. Because of the lackof gauge symmetry, the massive spin-2 perturbations con-tain a helicity-0 excitation. Moreover, it has been shownthat this helicity-0 excitation becomes a ghost if 0<M2 <2H2 [17,18]. This is exactly the same mass range for thediscrete mode in the � branch for Hrc > 1. Thus weexpect that there appears a ghost when Hrc > 1, that is,the universe with a positive cosmological constant isunstable.

IV. 4D EFFECTIVE ACTION FOR HELICITY-0EXCITATIONS

In summary, we expect that the helicity-0 excitation forthe spin-2 perturbations is a ghost for Hrc > 1 in the �branch. In order to confirm this result, we construct theeffective action for the helicity-0 excitations in the DGPmodel. It is convenient to take the metric as

ds2 � �1� 2Ayy�dy2 � 2N�y�Aydtdy

� N�y�2���1� 2A�dt2 � a2�1� 2R��ijdxidxj�;

where a�t� � exp�Ht�. In the absence of bulk matter, per-turbations are solved using a ‘‘master variable,’’ � [19]. Inthe special case of a de Sitter brane in a Minkowski bulk,the metric variables are written via the master variable � as

123511

A � �1

6a

�2�00 �

N0

N�0 �

1

N2���;

Ay �1

aN

�_�0 �

N0

N_��;

Ayy �1

6a

��00 � 2

N0

N�0 �

2

N2���;

R �1

6a

��00 �

N0

N�0 �

1

N2���;

(18)

where dot denotes a derivative with respect to t. In the bulk,the master equation for � can be solved by the separationof variables � �

Rdmgmk�t�fm�y�eikx and the equation of

motion in the bulk is given by

f00m � 2N0

Nf0m �

m2

N2 fm � 0;

�gmk � 3H �gmk �k2

a2 gmk � �m2gmk:

(19)

The junction conditions for � were derived in Ref. [20,21].For the vacuum brane, the boundary condition for � isgiven by

�0 �H��rc

1� 2Hrc�2H2 �m2�� � 0: (20)

We again find a tower of continuous massive modes start-ing from m2 � �9=4�H2, which are the KK modes for thespin-2 perturbations. In addition, there are two discretemodes. One is the mode with m2 � 2H2, which is thespin-0 perturbation and the other is the mode with m2 �m2d, which is the helicity-0 excitation of the spin-2 pertur-

bations. The 2nd order action for the master variable can becalculated by extending the result of Ref. [22]. Then wecan construct the 4D effective action for the discrete modeswith mass m2

i by integrating out with respect to y as

Si �k4N m2

i

6H�2

Zd4xa3 mk��4 �m

2i � mk; (21)

where

N m2i�

Hrc1� 2Hrc

�1

2�������������94�

m2i

H2

q ; (22)

and mk � a�3gmk. We should note that there is no mixingbetween two discrete modes for Hrc � 1. For the spin-0perturbation with m2 � 2H2, N m2

ibecomes

N 2H2 �1�Hrc1� 2Hrc

: (23)

N 2H2 is positive for H > 1=rc, so it is not a ghost. For thespin-2 perturbation with m2 � m2

d, N m2d

becomes

N m2d� �

�1�Hrc1� 2Hrc

��Hrc

3Hrc � 2

�: (24)

N m2d

is negative for H > 1=rc, which confirms that thespin-2 perturbations contain a ghost.

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non−ghost non−ghostghost

ghost

1/2 1 Hr c

χ

ϕ

µνhelicity−0

non−ghost

FIG. 1. Summary of the existence of the scalar ghost forvarious Hrc.

KAZUYA KOYAMA PHYSICAL REVIEW D 72, 123511 (2005)

V. SELF-ACCELERATING UNIVERSE

In a self-accelerating universe Hrc � 1, the spin-2graviton has mass m2 � 2H2, which coincides with themass of the brane fluctuation mode. For Hrc � 1, there isno longer a solution of the form Eq. (12), but there is asolution of the form h�� � A���x� � B���x� logN�y�where A�� and B�� are determined by ’ [23]. Then thereis a mixing between the spin-0 mode and the helicity-0excitation of the spin-2 graviton, which can give a ghost[23,24]. The derivation of the effective action is moresubtle in this case and a careful calculation of the effectiveaction is required.

However, we can argue the existence of the ghost in thefollowing way [24]. From the effective action for the branebending mode, it is clear that the brane bending modebecomes a ghost for 1=2<Hrc < 1 Eq. (23). On the otherhand, for Hrc > 1, the spin-2 graviton becomes a ghost(see Fig. 1). Thus, by continuity, it is likely to have a ghostforHrc � 1, in this case, from a mixing between the branebending mode and the spin-2 graviton. It should be men-tioned that this is consistent with the result obtained by theboundary effective action in Refs. [7,8], where a scalarmode is found to be a ghost if Hrc > 1=2.

VI. SUMMARY

In this paper, we studied the spectrum of gravitationalperturbations in the DGP model. In the self-accelerating

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branch, we showed that the spin-2 graviton has a discretemode with mass in the range 0<m2 � 2H2 for Hrc > 1where m2 � 2H2 for the self-accelerating universe Hrc �1 and m2 ! 0 for Hrc ! 1. Then the spin-2 gravitonacquires a helicity-0 excitation that is a ghost for Hrc >1, which confirms earlier results [7,8]. In addition, there isa normalisable brane fluctuation mode with mass m2 �2H2 that is not a ghost.

The self-accelerating universe is special in the sense thatthe spin-2 graviton has a mass m2 � 2H2, which is knownto be a special case in 4D PF massive gravity [17]. In theDGP model, there is a brane fluctuation mode with thesame mass, which can have a mixing with the spin-2graviton. We argued that this mixing presumably gives aghost in the self-accelerating universe. Recently, a newinstability is found in the limit of vanishing cosmologicalconstant in the analysis of the gravitational shock wavefields generated by a source on the brane [25]. This may bea signal of the appearance of the physical ghost in thislimit. In order to verify the existence of the ghost in thisspecial case rigorously, a more careful calculation of theeffective action is required and this will be presented in afuture publication [26].

ACKNOWLEDGMENTS

We would like to thank C. Deffayet, N. Kaloper, K.Koyama, D. Langlois, R. Maartens, and M. Sasaki fordiscussions. We would also like to thank D. S. Gorbunovand S. M. Sibiryakov for pointing out the existence of asolution for the spin-0 mode in the self-accelerating uni-verse and R. Rattazzi for pointing out the possibility ofhaving a ghost from a mixing between the brane fluctuationmode and the spin-2 graviton in a self-accelerating uni-verse. The author benefitted from discussions during theconference ‘‘The Next Chapter in Einstein’s Legacy’’ atthe Yukawa institute, Kyoto and the subsequent workshop.This work was supported by JSPS and PPARC.

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