Spectrum from the warped compactifications with the de Sitter universe

We discuss the spectrum of the tensor metric perturbations and the stability of warped compactifications with the de Sitter spacetime in the higher-dimensional gravity. The spacetime structure is given in terms of the warped product of the non-compact direction, the spherical internal dimensions and the four-dimensional de Sitter spacetime. To realize a finite bulk volume, we construct the brane world model, using the cut-copy-paste method. Then, we compactify the spherical directions on the brane. In any case, we show the existence of the massless zero mode and the mass gap of it with massive Kaluza-Klein modes. Although the brane involves the spherical dimensions, no light massive mode is excited. We also investigate the scalar perturbations, and show that the model is unstable due to the existence of a tachyonic bound state, which seems to have the universal negative mass square, irrespective of the number of spacetime dimensions.Comment: Journal version (JHEP

Black String Perturbations in RS1 Model

We present a general formalism for black string perturbations in Randall-Sundrum 1 model (RS1). First, we derive the master equation for the electric part of the Weyl tensor $E_{\mu\nu}$. Solving the master equation using the gradient expansion method, we give the effective Teukolsky equation on the brane at low energy. It is useful to estimate gravitational waves emitted by perturbed rotating black strings. We also argue the effect of the Gregory-Laflamme instability on the brane using our formalism.Comment: 14 pages, Based on a talk presented at ACRGR4, the 4th Australasian Conference on General Relativity and Gravitation, Monash University, Melbourne, January 2004. To appear in the proceedings, in General Relativity and Gravitatio

Effective theory for close limit of two branes

We discuss the effective theory for the close limit of two branes in a covariant way. To do so we solve the five dimensional Einstein equation along the direction of the extra dimension. Using the Taylor expansion we solve the bulk spacetimes and derive the effective theory describing the close limit. We also discuss the radion dynamics and braneworld black holes for the close limit in our formulation.Comment: 6 pages, a version to be published in Phy.Rev.

Gravity, Stability and Energy Conservation on the Randall-Sundrum Brane-World

We carefully investigate the gravitational perturbation of the Randall-Sundrum (RS) single brane-world solution [hep-th/9906064], based on a covariant curvature tensor formalism recently developed by us. Using this curvature formalism, it is known that the `electric' part of the 5-dimensional Weyl tensor, denoted by $E_{\mu\nu}$, gives the leading order correction to the conventional Einstein equations on the brane. We consider the general solution of the perturbation equations for the 5-dimensional Weyl tensor caused by the matter fluctuations on the brane. By analyzing its asymptotic behaviour in the direction of the 5th dimension, we find the curvature invariant diverges as we approach the Cauchy horizon. However, in the limit of asymptotic future in the vicinity of the Cauchy horizon, the curvature invariant falls off fast enough to render the divergence harmless to the brane-world. We also obtain the asymptotic behavior of $E_{\mu\nu}$ on the brane at spatial infinity, assuming the matter perturbation is localized. We find it falls off sufficiently fast and will not affect the conserved quantities at spatial infinity. This indicates strongly that the usual conservation law, such as the ADM energy conservation, holds on the brane as far as asymptotically flat spacetimes are concerned.Comment: 10 pages, references adde

Scalar perturbations from brane-world inflation

We investigate the scalar metric perturbations about a de Sitter brane universe in a 5-dimensional anti de Sitter bulk. We compare the master-variable formalism, describing metric perturbations in a 5-dimensional longitudinal gauge, with results in a Gaussian normal gauge. For a vacuum brane (with constant brane tension) there is a continuum of normalizable Kaluza-Klein modes, with m>3H/2, which remain in the vacuum state. A light radion mode, with m=\sqrt{2}H, satisfies the boundary conditions for two branes but is not normalizable in the single-brane case. When matter is introduced (as a test field) on the brane, this mode, together with the zero-mode and an infinite ladder of discrete tachyonic modes, become normalizable. However, the boundary condition requires the self-consistent 4-dimensional evolution of scalar field perturbations on the brane and the dangerous growing modes are not excited. These normalizable discrete modes introduce corrections at first-order to the scalar field perturbations computed in a slow-roll expansion. On super-Hubble scales, the correction is smaller than slow-roll corrections to the de Sitter background. However on small scales the corrections can become significant.Comment: 15 page

Geometry and cosmological perturbations in the bulk inflaton model

We consider a braneworld inflation model driven by the dynamics of a scalar field living in the 5-dimensional bulk, the so-called ``bulk inflaton model'', and investigate the geometry in the bulk and large scale cosmological perturbations on the brane. The bulk gravitational effects on the brane are described by a projection of the 5-dimensional Weyl tensor, which we denote by $E_{\mu\nu}$. Focusing on a tachionic potential model, we take a perturbative approach in the anti-de Sitter (AdS$_5$) background with a single de Sitter brane. We first formulate the evolution equations for $E_{\mu\nu}$ in the bulk. Next, applying them to the case of a spatially homogeneous brane, we obtain two different integral expressions for $E_{\mu\nu}$. One of them reduces to the expression obtained previously when evaluated on the brane. The other is a new expression that may be useful for analyzing the bulk geometry. Then we consider superhorizon scale cosmological perturbations and evaluate the bulk effects onto the brane. In the limit $H^2\ell^2\ll1$, where $H$ is the Hubble parameter on the brane and $\ell$ is the bulk curvature radius, we find that the effective theory on the brane is identical to the 4-dimensional Einstein-scalar theory with a simple rescaling of the potential even under the presence of inhomogeneities. % atleast on super-Hubble horizon scales. In particular, it is found that the anticipated non-trivial bulk effect due to the spatially anisotropic part of $E_{\mu\nu}$ may appear only at %second order in the low energy expansion, i.e., at $O(H^4\ell^4)$.Comment: 21 pages including 6 pages for several appendixes, no figure

Strong Brane Gravity and the Radion at Low Energies

For the 2-brane Randall-Sundrum model, we calculate the bulk geometry for strong gravity, in the low matter density regime, for slowly varying matter sources. This is relevant for astrophysical or cosmological applications. The warped compactification means the radion can not be written as a homogeneous mode in the orbifold coordinate, and we introduce it by extending the coordinate patch approach of the linear theory to the non-linear case. The negative tension brane is taken to be in vacuum. For conformally invariant matter on the positive tension brane, we solve the bulk geometry as a derivative expansion, formally summing the `Kaluza-Klein' contributions to all orders. For general matter we compute the Einstein equations to leading order, finding a scalar-tensor theory with $\omega(\Psi) \propto \Psi / (1 - \Psi)$, and geometrically interpret the radion. We comment that this radion scalar may become large in the context of strong gravity with low density matter. Equations of state allowing $(\rho - 3 P)$ to be negative, can exhibit behavior where the matter decreases the distance between the 2 branes, which we illustrate numerically for static star solutions using an incompressible fluid. For increasing stellar density, the branes become close before the upper mass limit, but after violation of the dominant energy condition. This raises the interesting question of whether astrophysically reasonable matter, and initial data, could cause branes to collide at low energy, such as in dynamical collapse.Comment: 24 pages, 3 figure

Thick de Sitter 3-Branes, Dynamic Black Holes and Localization of Gravity

The embedding of a thick de Sitter 3-brane into a five-dimensional bulk is studied, assuming a scalar field with potential is present in the bulk. A class of solutions is found in closed form that can represent a thick de Sitter 3-brane interpolating either between two dynamical black holes with a $R \times S_{4}$ topology or between two Rindler-like spacetimes with a $R_{2}\times S_{3}$ topology. The gravitational field is localized in a small region near the center of the 3-brane. The analysis of graviton fluctuations shows that a zero mode exists and separates itself from a set of continuous modes by a mass gap. The existence of such a mass gap is shown to be universal. The scalar perturbations are also studied and shown to be stable.Comment: the study of scalar perturbations and some relevant references have been added. The most used definition for mass in de Sitter space has been adopte

Wave function of the radion in the brane background with a massless scalar field and a self-tuning problem

We consider flat solutions in the brane background with a massless scalar field appearing in 5D $H^2_{MNPQ}$. Since there exist bulk singularities or arises the divergent 4D Planck mass, we should introduce a compact extra dimension, the size of which is then fixed by brane tension(s) and a bulk cosmological constant. Inspecting scalar perturbations around the flat solutions, we find that the flat solutions are stable vacua from the positive mass spectrum of radion. We show that the massless radion mode is projected out by the boundary condition arising in cutting off the extra dimension. Thus, the fixed extra dimension is not alterable, which is not useful toward a self-tuning of the cosmological constant.Comment: Latex file of 18 pages including 1 eps figur

Can Inflating Braneworlds be Stabilized?

We investigate scalar perturbations from inflation in braneworld cosmologies with extra dimensions. For this we calculate scalar metric fluctuations around five dimensional warped geometry with four dimensional de Sitter slices. The background metric is determined self-consistently by the (arbitrary) bulk scalar field potential, supplemented by the boundary conditions at both orbifold branes. Assuming that the inflating branes are stabilized (by the brane scalar field potentials), we estimate the lowest eigenvalue of the scalar fluctuations - the radion mass. In the limit of flat branes, we reproduce well known estimates of the positive radion mass for stabilized branes. Surprisingly, however, we found that for de Sitter (inflating) branes the square of the radion mass is typically negative, which leads to a strong tachyonic instability. Thus, parameters of stabilized inflating braneworlds must be constrained to avoid this tachyonic instability. Instability of "stabilized" de Sitter branes is confirmed by the BraneCode numerical calculations in the accompanying paper hep-th/0309001. If the model's parameters are such that the radion mass is smaller than the Hubble parameter, we encounter a new mechanism of generation of primordial scalar fluctuations, which have a scale free spectrum and acceptable amplitude.Comment: 7 pages, RevTeX 4.