# Brane gravity, higher derivative terms and non-locality

###### Abstract

In brane world scenarios with a bulk scalar field between two branes it is known that -dimensional Einstein gravity is restored at low energies on either brane. By using a gauge-invariant gravitational and scalar perturbation formalism we extend the theory of weak gravity in the brane world scenarios to higher energies, or shorter distances. We argue that weak gravity on either brane is indistinguishable from -dimensional higher derivative gravity, provided that the inter-brane distance (radion) is stabilized, that the background bulk scalar field is changing near the branes and that the background bulk geometry near the branes is warped. This argument holds for a general conformal transformation to a frame in which matter on the branes is minimally coupled to the metric. In particular, Newton’s constant and the coefficients of curvature-squared terms in the -dimensional effective action are determined up to an ambiguity of adding a Gauss-Bonnet topological term. In other words, we provide the brane-world realization of the so called -model without utilizing a quantum theory. We discuss the appearance of composite spin- and spin- fields in addition to the graviton on the brane and point out a possibility that the spin- field may play the role of an effective inflaton to drive brane-world inflation. Finally, we conjecture that the sequence of higher derivative terms is an infinite series and, thus, indicates non-locality in the brane world scenarios.

###### pacs:

PACS numbers: 04.50.+h; 98.80.Cq; 12.10.-g; 11.25.Mj^{†}

^{†}preprint: HUTP-01/A064,hep-th/0112205

## I Introduction

In the recent development of string/M theory [1], branes have been playing many important roles. The idea that our universe is a brane in a higher dimensional spacetime has been attracting a great deal of interest [2, 3, 4, 5]. Although the idea of the brane world had arisen at a phenomenological level already in 1983 [6], it is perhaps the discovery of the duality between M-theory and heterotic superstring theory by Horava and Witten [7] that made it more attractive. It actually gives the brane world idea a theoretical background: by compactifying six dimensions in the -dimensional theory, our -dimensional universe may be realized as a hypersurface in dimensions at one of the fixed points of an compactification. After compactification of dimensions by a Calabi-Yau manifold, the -dimensional effective theory can be obtained, see e.g. [8].

Randall and Sundrum proposed two similar but distinct phenomenological brane world scenarios [4, 5]. In the scenario the -dimensional spacetime is compactified on and all matter fields are assumed to be confined on branes at fixed points of the so that the bulk, or the spacetime region between two fixed points, is described by pure Einstein gravity with a negative cosmological constant. In the second scenario the fifth dimension is infinite but still the symmetry is imposed. In both scenarios the existence of the branes and the bulk cosmological constant makes the bulk geometry curved, or warped. There are generalizations of their scenarios with a scalar field between two branes [9, 10]. In these generalized warped brane-world scenarios, the scalar field was introduced to stabilize a modulus called the radion, which represents a separation between the two branes.

In the brane world scenarios with or without a scalar field in a warped bulk geometry, weak gravity in a static background has been extensively investigated [11, 12, 13, 14, 15, 16]. It is known that weak gravity in the scenario without a bulk scalar between two branes is not Einstein but a Brans-Dicke theory at low energies. On the other hand, in scenarios with a bulk scalar field between two branes -dimensional Einstein gravity is restored at low energies. It is believed that the validity of Einstein gravity breaks down at a certain energy scale which can be much lower than -dimensional Planck energy.

Hence, it seems natural to ask ’what does gravity in brane worlds look like at high energies or at short distances?’ In other words, ’how does the -dimensional description break down?’

In this paper we investigate weak gravity in brane world scenarios with a bulk scalar field between two branes at higher energies. For this purpose we use the gauge-invariant perturbation formalism developed in ref. [16]. In this formalism all quantities and equations are Fourier transformed with respect to the -dimensional coordinates and classified into scalar, vector and tensor perturbations so that the problem is reduced to a set of purely -dimensional problems. We also adopt expansion in a parameter , where is a characteristic length scale of the model and is the -dimensional momentum (or the Fourier parameter). In the lowest order in , -dimensional Einstein gravity is restored on either brane [16]. In the next order it is shown that gravity on either brane is indistinguishable from a higher derivative gravity whose action includes the Einstein term and curvature-squared terms. Equipped with the result for this order, we conjecture that in the order , gravity on either brane is indistinguishable from a higher derivative gravity whose action includes terms of up to the -th power of curvature tensors. Noting that the expansion in is in principle an infinite series, this conjecture indicates that gravity on either brane is non-local at high energies even at the linearized level. This explains how the -dimensional description breaks down at high energies. Physically, the non-locality is due to gravitational and scalar waves in the bulk.

This paper is organized as follows. In section II we summarize the basic equations by reviewing the formulation given in ref. [16]. In section III we perform the low energy expansion to investigate the system. In section IV we review linear perturbations in -dimensional higher derivative gravity so as to compare it with gravity in the brane world. In section V we discuss some physical implications. Finally, section VI is devoted to a summary of results.

## Ii Basic equations

The model and basic equations we shall investigate in this paper are exactly the same as those in ref. [16]. We now summarize them briefly.

### ii.1 Model description and background

We consider a -dimensional spacetime of the topology , where represents -dimensional spacetime. We denote two timelike hypersurfaces corresponding to fixed points of the compactification by . Each hypersurface can be considered as the world volume of a -brane. In order to describe we use the parametric equations

(1) |

where are -dimensional coordinates in and each denotes four parameters . The four parameters play the role of -dimensional coordinates on each hypersurface. It is notable that the -dimensional coordinates are not necessarily a part of the -dimensional coordinates. Actually, in the following we shall consider a -dimensional gauge transformation (a -gauge transformation) on each brane and a -dimensional gauge transformation (a -gauge transformation) in the bulk independently, where we call the -dimensional region between the bulk (and we shall denote it by ). In particular, a quantity invariant under the latter (a -gauge invariant variable) is not necessarily invariant under the former (-gauge invariant).

We consider a theory described by the action

(2) |

where is the -dimensional Einstein-Hilbert action

(3) |

is the action of a scalar field

(4) |

and is the action of matter fields confined on the branes

(5) |

Here, and represent the pullback of and the induced metric on , and is the physical metric on , which is not necessarily equivalent to . We assume that the physical metric is related to the induced metric by a conformal transformation depending on :

(6) |

where is a function of , respectively. As shown in ref. [17] the variational principle based on the action (2) gives the correct set of equations of motion, including variations of , and . It is essential that the region of integration in the Einstein action (4) is not but : the integration across gives the so called Gibons-Hawking term correctly.

We consider general perturbations around a background with -dimensional Poincare symmetry:

(7) |

where the background is given by

(8) |

() represent first four of -dimensional coordinates () in , represents the fifth coordinate , and () are constants. Here, is the physical surface energy momentum tensor defined by

(9) |

and we have redefined and so that vanishes. Hereafter, we assume that the brane at is our brane and that there is no excitation of matter on the other brane ().

The equations of motion for the background are as follows.

(10) |

and

(11) |

where dots denote derivative with respect to and . Here, we assumed that and that the bulk is the region . The induced metric and the physical metric on for the background are

(12) |

### ii.2 Gauge-invariant variables

Let us now construct gauge-invariant variables from the metric perturbation , the scalar field perturbation , the brane fluctuation and the matter perturbation . There are actually two types of gauge-invariant variables as there are two types of gauge-transformations: the -dimensional gauge transformation in the bulk (-gauge transformation)

(13) |

and the -dimensional gauge transformation on each brane (-gauge transformation)

(14) |

As pointed out in ref. [37], these two kinds of gauge-transformations are independent.

For the purpose of construction of gauge-invariant variables we expand all perturbations by harmonics in -dimensional Minkowski spacetime. This strategy is convenient since the background has -dimensional Poincare symmetry and the induced (and physical) metric on each brane is -dimensional Minkowski metric. In appendix A we define scalar harmonics , vector harmonics and tensor harmonics . By using those harmonics we can expand all perturbations as

(15) |

and

(16) |

where we omitted dependence of harmonics and the corresponding coefficients on the -dimensional momentum and the integration with respect to . The -dependent Fourier coefficients , , , and are functions of the fifth coordinate only, and the other coefficients , , and are constants.

Now we can analyze -gauge transformation of the coefficients of the harmonic expansion and construct -gauge-invariant variables, or those linear combinations of perturbations that are invariant under the -gauge transformation. The result is

(17) |

where . They form a maximal set of independent -gauge invariant variables constructed from the metric perturbation , the scalar field perturbation and the brane fluctuation . (The matter perturbation is not included here since it is a -dimensional object.)

We can also analyze -gauge transformation and construct -gauge-invariant variables, or those linear combinations of perturbations that are invariant under the -gauge transformation. For this purpose we first need to obtain expressions of various -dimensional quantities in terms of the -dimensional quantities , and . What we need are the physical metric defined by (6), the extrinsic curvature , the pull back of the scalar field , and the normal derivative of on :

(18) |

where

(19) |

and represents a -dimensional Lie derivative. Using the harmonic expansions (15), we can obtain the corresponding harmonic expansion of , , and . The result is

(20) |

where is defined by

(21) |

and -dependent Fourier coefficients are

(22) |

Here, the right hand sides of (22) are evaluated at
, respectively, and have been written in terms of -gauge
invariant variables ^{1}^{1}1The reason why they can be expressed in
terms of -gauge-invariant variables only is that they by themselves
are -gauge-invariant [33]. This fact illustrates
that -gauge transformation is not a part of -gauge transformation
and that these two kinds of gauge transformations are independent.. The
matter perturbation on each brane can also
be expanded by harmonics as

(23) |

We can now analyze -gauge transformation of coefficients the harmonic expansion and construct the following -gauge invariant variables from the physical metric perturbation .

(24) |

It is easily shown that , , and are invariant under the -gauge transformation. Moreover, they are at the same time -gauge invariant since all except for are written in terms of -gauge invariant variables and is a -dimensional object. Hence, we have the set (, , , , , ) of doubly-gauge invariant variables.

From the point of view of observers on each brane, all observable quantities must be doubly-gauge invariant. However, all doubly-gauge invariant variables are not necessarily observable quantities. They can observe physical metric perturbation (, ) and matter perturbation only. The remaining doubly-gauge invariant variables (, , ) shall be used to write down junction conditions of -dimensional quantities in a doubly-gauge invariant way.

Our remaining task in this section is to give -gauge invariant equations in the bulk and doubly-gauge invariant junction conditions on each brane. Our final aim in this paper is to seek doubly-gauge invariant equations governing the physical metric perturbations and matter perturbations on our brane and to compare the resulting equations with the corresponding equations in -dimensional higher derivative gravity.

Since there are many coefficients in the above harmonic expansions, let us divide these into three classes. The first class is the scalar perturbations and consists of coefficients of , , and . The second is the vector perturbations and consists of coefficients of and . The last is the tensor perturbations and consists of coefficients of . Perturbations in different classes are decoupled from each other at the linearized level. Hence, in the following we analyze perturbations in each class separately. Decomposition into scalar, vector and tensor modes is commonly used in cosmology. However, usually in cosmology we use scalar, vector and tensor representations of the isometry group related to the symmetry of -dimensional space. Meanwhile, here we will use scalar, vector and tensor representations of the isometry group of -dimensional space-time.

In ref. [16] it was shown that vector type perturbations vanish unless matter fields on the hidden brane are excited. In this paper we assume that there is no matter excitation on the hidden brane and, thus, we shall consider scalar and tensor type perturbations only.

### ii.3 Scalar perturbations

For scalar perturbations, we have three -gauge invariant variables from metric perturbation and scalar field perturbation in the bulk: , and . The first two are from metric perturbation and the last one is from scalar field perturbation. The Einstein equation leads to two relations among them

(25) |

and a wave equation

(26) |

where a dot denotes derivative with respect to and is the -dimensional momentum in the coordinate . Throughout this paper we consider modes with for scalar perturbations since a scalar mode with preserves -dimensional Poincare symmetry and, thus, represents just a change of the background within the ansatz (8). Of course we shall consider modes with as long as .

The boundary condition at , respectively, is given by the junction condition for the scalar field as

(27) |

where and are doubly gauge
invariant variables constructed from matter on and
perturbation of the position of , respectively, and
satisfy the following equations derived from the perturbed Israel’s
junction condition ^{2}^{2}2For a mode with we do not have
the second equation of (28) since there is no
tensor harmonics of the type for . See appendix
A for definition and properties of harmonics..

(28) |

Here, , and is another doubly gauge invariant variable constructed from matter on . The first of (28) is nothing but the conservation equation of matter stress energy tensor on each brane. The second equation is the -component of the Israel’s junction condition and relates the perturbation of the brane position and the matter perturbation. The boundary condition (27) at , respectively, can be rewritten to the following form by eliminating , and , and using the wave equation (26).

(29) |

where

(30) |

Finally, the doubly gauge invariant perturbation of the physical metric is expressed as

(31) |

In the next section we shall assume that to show the recovery of higher derivative gravity on a brane. In this case, the expression of can be rewritten to the following form by eliminating and , and using the boundary condition (29).

(32) |

Hereafter, we consider as our brane and as the hidden brane, and assume that there is no matter excitations on the hidden brane. Hence, we put .

### ii.4 Tensor perturbations

For tensor perturbations we have only one -gauge invariant variable constructed from metric perturbation in the bulk: . The Einstein equation leads to the wave equation

(33) |

where is the -dimensional momentum in the coordinate .

The Israel junction condition leads to the following boundary condition at , respectively.

(34) |

where is a doubly gauge invariant variable constructed from matter on .

Finally, the doubly gauge invariant perturbation of the physical metric is expressed as

(35) |

Hereafter, since we assumed that there is no matter excitations on the hidden brane, we put .

## Iii Low energy expansion

As already stated in the third-to-the-last paragraph of subsection II.2, our aim in this paper is to seek doubly-gauge invariant equations governing the physical metric perturbations and matter perturbations on our brane and to compare the resulting equations with the corresponding equations in -dimensional higher derivative gravity. Since we have only two gauge-invariant metric perturbations and , and essentially two gauge-invariant matter perturbations and on our brane ( is related to by the conservation equation), we expect the following form of the equations on the brane.

(36) |

where and are functions of
and
is the inverse of the
unperturbed physical metric . The reason why we
expect the linear dependence of metric perturbations on matter
perturbations is that boundary conditions summarized in the previous
section are linear in the matter perturbations. The reason why
was put in front of
is that we expect -dimensional gravitons on our
brane (non-vanishing with
and ). What
is important here is that the functions completely
characterize the effective theory of weak gravity on our
brane ^{3}^{3}3If one likes, one can restore gauge fixed
equations for any gauge choices from the functions only. For
example, see (71).. Since we are dealing with
gauge-invariant variables only, there is no ambiguity of gauge freedom
when we compare the result with the corresponding equations in
-dimensional higher derivative gravity. Namely, we only have to
compare functions of
with the corresponding functions of momentum squared in the Fourier
transformed, linearized -dimensional higher derivative gravity.

In this section we expand the basic equations summarized in the previous section by the parameter and solve them iteratively, where is the characteristic length scale of the model which we shall determine by comparing the results of order and . The purpose of the -expansion is to analyze the behavior of the functions and near . Namely, we shall seek first few coefficients () of the expansion

(37) |

Since the -dimensional physical energy scale on is given by , the expansion in is nothing but the low energy expansion. Hence, the first few coefficients () of the expansion (37) determine the low energy behavior of weak gravity on our brane. We expect that give -dimensional Einstein gravity, that give curvature-squared corrections to the Einstein gravity, and so on. The length scale gives the energy scale below which we can trust the -expansion. Nonetheless, we can defer the determination of until we will obtain the results of order since can be eliminated from all formal results in each order of the -expansion. Meanwhile, we shall keep it in intermediate calculations in order to make the expansion parameter dimensionless.

First, by expanding and as

(38) |

we can solve the wave equations (26) and (33) order by order. The result is

(39) | |||||

with , where and are constants.

Next, let us analyze the boundary condition. For this purpose, we expand