In this work, we investigate the gravitational resonances in various $f(T)$-brane models with the warp factor $\text{e}^{A(y)}=\tanh\big(k(y+b)\big)-\tanh\big(k(y-b)\big)$. For three kinds of $f(T)$, we give the solutions to the system. Besides, we consider the tensor perturbation of vielbein and obtain the effective potentials by the Kaluza-Klein (KK) decomposition. Then, we analyze what kind of effective potential can produce the gravitational resonances. Effects of different parameters on the gravitational resonances are analysed. The lifetimes of the resonances could be long enough as the age of our universe in some ranges of the parameters. This indicates that the gravitational resonances might be considered as one of the candidates of dark matter. Combining the current experimental observations, we constrain the parameters for these brane models.
In this paper, we investigate various $f(R)$-brane models and compare their gravitational resonance structures with the corresponding general relativity (GR)-branes. {Starting from some known GR-brane solutions}, we derive thick $f(R)$-brane solutions such that the metric, scalar field, and scalar potential coincide with those of the corresponding GR-branes. {We find that for branes generated by a single or several canonical scalar fields, there is no obvious distinction between the GR-branes and corresponding $f(R)$-branes in terms of gravitational resonance structure.} Then we discuss the br…
We investigate the braneworld gravity starting from the non-conservative gravitational field equations in a five-dimensional bulk. The approach is based on the Gauss-Codazzi formalism along with the study of the braneworld consistency conditions. The effective gravitational equations on the brane are obtained and the constraint leading to a brane energy-momentum conservation is analyzed.
Lorentz violation in a brane-world scenario is presented and used to obtain a relationship between the speed of gravitational waves in the bulk and that on the brane. Lorentz violating effects would manifest themselves in gravitational waves travelling with a greater speed in the bulk than on the brane and this effect is independent of the signature of the extra dimension.
We consider intersecting black branes with strong gravitational waves propagating along their worldvolume in the context of supergravity theories. Both near-horizon and space-filling gravitational wave modes are included in our ansatz. The equations of motion (originally, partial differential equations) are shown to reduce to ordinary differential equations, which include a Toda-like system. For special arrangements of intersecting black branes, the Toda-like system becomes integrable, permitting a more thorough analysis of the gravitational equations of motion.
In this work, we investigate numerical evolution of massive Kaluza-Klein (KK) modes of a scalar field in a thick brane. We derive the Klein-Gordon equation in five dimensional spacetime, and obtain the evolution equation and the Schrödinger-like equation. With the resonances of the scalar KK modes as the initial data, the scalar field is evolved with the maximally dissipative boundary condition. The results show that there are scalar KK resonant particles with long life on the brane, which indicates that these resonances might be viewed as one of the candidates for dark matter.
In this paper, by presenting the potentials of Kaluza-Klein (KK) modes in the corresponding Schrödinger equations, we investigate the localization and resonances of gravity and fermion on the symmetric and asymmetric Bloch {branes}. We find that the localization properties of zero modes for gravity and fermion in the symmetric brane case are the same, whereas, for the asymmetric brane case, the fermion zero mode is localized on one of the sub-branes, while the gravity zero mode is localized on another sub-brane. The spectra of the gravity and the left- or right-handed fermion are composed of a…
We introduce a new kind of space-filling brane, which we term "G-brane" because its action is a descendant of the gravitational action. The G-brane may be thought of as the remanent of the gravitational field when the propagating gravitons are removed. The G-brane is different from the Dirac or Nambu space-filling branes. Its properties in any spacetime dimension D are exhibited. When the spacetime dimension D is greater than or equal to three, the G-brane does not possess propagating degrees of freedom, just as the Dirac or Nambu branes. For D=3 the G-brane yields a reformulation of gravitati…
The viability of achieving gravitational consistent braneworld models in the framework of a f(R) theory of gravity is investigated. After a careful generalization of the usual junction conditions encompassing the embedding of the 3-brane into a f(R) bulk, we provide a prescription giving the necessary constraints in order to implement the projected second order effective field equations on the brane.
Stability and localization of the gravitational perturbations for a special brane system in Eddington-inspired Born-Infeld (EiBI) gravity were studied in [Phys. Rev. D 85, 124053 (2012)]. In this paper, we show that the gravitational perturbations for a general brane system are stable, the four-dimensional graviton (massless KK graviton) can be localized on the brane, and the mass spectrum of massive KK gravitons are gapless and continuous. Two models are constructed as examples. In the first model, which is a generalization of [Phys. Rev. D 85, 124053 (2012)], the brane has no inner structure…
We consider the Cauchy problem of the gravitational wave with the initial data distributed only around the brane in the one brane model of Randall and Sundrum, and examine its behavior as t\to\infty. Then we find its leading behavior is t^{-6} unlike an ordinary flat 5-dimensional space-time. Such a signal shows that the Huygens principle is violated on the 4-dimensional brane world even asymptotically and also shows the difference between compact and non-compact brane worlds. Some comments are also given related to AdS/CFT correspondence.
We investigate in detail gravitational waves in an Schwarzschild-anti-de Sitter bulk spacetime surrounded by an Einstein static brane with generic matter content. Such a model provides a useful analogy to braneworld cosmology at various stages of its evolution, and generalizes our previous work [gr-qc/0504023] on pure tension Einstein-static branes. We find that the behaviour of tensor-mode perturbations is completely dominated by quasi-normal modes, and we use a variety of numeric and analytic techniques to find the frequencies and lifetimes of these excitations. The parameter space governing…
Consistent Lagrangian description of lightlike p-branes (LL-branes) is presented in two equivalent forms - a Polyakov-type formulation and a dual to it Nambu-Goto-type formulation. An important characteristic feature of the LL-branes is that the brane tension appears as a non-trivial additional dynamical degree of freedom. Next, properties of p=2 LL-brane dynamics (as a test brane) in D=4 Kerr or Kerr-Newman gravitational backgrounds are discussed in some detail. It is shown that the LL-brane automatically positions itself on the horizon and rotates along with the same angular velocity. Finall…
In this work, we investigate gravitational resonances in both single and double mimetic thick branes, which can provide a new way to detect the extra dimension. For the single brane model, we apply the relative probability proposed in [Phys. Rev. D. 80 (2009) 065019]. For the double brane model, we investigate the resonances quasi-localized on the double brane, on the sub-branes and between the sub-branes, respectively. To investigate the resonances quasi-localized on the double brane, we introduce two different definitions of the relative probability and find that the corresponding mass spect…
We present effective gravitational equations at low energies in a $Z_2$-symmetric braneworld with the Gauss-Bonnet term. Our derivation is based on the geometrical projection approach, and we solve iteratively the bulk geometry using the gradient expansion scheme. Although the original field equations are quite complicated due to the presence of the Gauss-Bonnet term, our final result clearly has the form of the Einstein equations plus correction terms, which is simple enough to handle. As an application, we consider homogeneous and isotropic cosmology on the brane. We also comment on the holo…
We present the covariant gravitational equations to describe a four-dimensional brane world in the case with the Gauss-Bonnet term in a bulk spacetime, assuming that gravity is confined on the $Z_2$ symmetric brane. It contains some components of five-dimensional Weyl curvature ($E_{μν}$) which describes all effects from the bulk spacetime just as in the case of the Randall-Sundrum second model. Applying this formalism to cosmology, we derive the generalized Friedmann equation and calculate the Weyl curvature term, which is directly obtained from a black hole solution.
We consider the propagation of gravitational waves in six dimensions induced by sources living on 3-branes in the context of a recent exact solution hep-th/0506050. The brane geometries are de Sitter and the bulk is a warped geometry supported by a positive cosmological constant as well as a 2-form flux. We show that at low energies ordinary gravity is reproduced, and explicitly compute the leading corrections from six dimensional effects. After regulating the brane we find a logarithmic dependence on the cutoff scale of brane physics even for modes whose frequency is much less than this energ…
Tensor resonances in teleparallel Gauss-Bonnet branes
Federal University of Ceará (UFC)（塞阿拉联邦大学）；Secretaria da Educação do Ceará (SEDUC), Coordenadoria Regional de Desenvolvimento da Educação (CREDE 9)（塞阿拉州教育厅，区域教育发展协调局（第9区））；Universidade Federal do ABC (UFABC)（ABC联邦大学）
We study gravitational perturbations in the Randall-Sundrum two-brane background with scalar-curvature terms in the action for the branes, allowing for positive as well as negative bulk gravitational constant. In the zero-mode approximation, we derive the linearized gravitational equations, which have the same form as in the original Randall-Sundrum model but with different expressions for the effective physical constants. We develop a generic method for finding tachyonic modes in the theory, which, in the model under consideration, may exist only if the bulk gravitational constant is negative…
We derive a low-energy effective theory for gravity with anti-D branes, which are essential to get de Sitter solutions in the type IIB string warped compactification, by taking account of gravitational backreactions of anti-D branes. In order to see the effects of the self-gravity of anti-D branes, a simplified model is studied where a 5-dimensional anti-de Sitter ({\it AdS}) spacetime is realized by the bulk cosmological constant and the 5-form flux, and anti-D branes are coupled to the 5-form field by Chern-Simon terms. The {\it AdS} spacetime is truncated by introducing UV and IR cut-off br…
We consider spherically symmetric solutions within the context of brane-world theory without mirror symmetry or any form of junction conditions. For a constant curvature bulk, we obtain the modified Tolman-Oppenheimer-Volkoff (TOV) interior solutions in two cases where one is matched to a schwarzschild-de Sitter exterior while the other is consistent with an exterior solution whose structure can be used to explain the galaxy rotation curves without postulating dark matter. We also find the upper bound to the mass of a static brane-world star and show that the influence of the bulk effects on t…
We study information theoretic properties of planar black hole microstates in 2 + 1 dimensional asymptotically anti-de Sitter spacetime, modeled by black holes with an end-of-the-world brane behind the horizon. The von Neumann entropy of sufficiently large subregions in the dual CFT exhibits a time-dependent phase, which from a doubly-holographic perspective corresponds to the appearance of quantum extremal islands in the brane description. Considering the case where dilaton gravity is added to the brane, we show that tuning the associated couplings affects the propagation of information in th…
It is known that there are two mechanisms for localizing a bulk fermion on a brane, one is the well-known Yukawa coupling and the other is the new coupling proposed in [Phys. Rev. D 89, 086001 (2014)]. In this paper, we investigate localization and resonance spectrum of a bulk fermion on the same branes with the two localization mechanisms. It is found that both the two mechanisms can result in a volcano-like effective potential of the fermion Kaluza-Klein modes. The left-chiral fermion zero mode can be localized on the brane and there exist some discrete massive fermion Kaluza-Klein modes tha…
We consider some classes of solutions of the static, spherically symmetric gravitational field equations in the vacuum in the brane world scenario, in which our Universe is a three-brane embedded in a higher dimensional space-time. The vacuum field equations on the brane are reduced to a system of two ordinary differential equations, which describe all the geometric properties of the vacuum as functions of the dark pressure and dark radiation terms (the projections of the Weyl curvature of the bulk, generating non-local brane stresses). Several classes of exact solutions of the vacuum gravitat…
In this paper, we study the structure and gravitational resonances of thick branes generated by a mimetic scalar field in $f(R)$ gravity. We obtain several typical thick brane solutions for $f(R)=R+αR^2$. To study their stability, we analyze the tensor perturbation of the metric. It is shown that any thick brane model with $df/dR>0$ is stable and the graviton zero mode can be localized on the brane for each solution, which indicates that the four-dimensional Newtonian gravity can be restored. The effect of the parameter $α$ on the gravitational resonances is studied. As a brane splits into mul…
We characterize the worldvolume theories on symmetric D-branes in a six-dimensional Cahen-Wallach pp-wave supported by a constant Neveu-Schwarz three-form flux. We find a class of flat noncommutative euclidean D3-branes analogous to branes in a constant magnetic field, as well as curved noncommutative lorentzian D3-branes analogous to branes in an electric background. In the former case the noncommutative field theory on the branes is constructed from first principles, related to dynamics of fuzzy spheres in the worldvolumes, and used to analyse the flat space limits of the string theory. The …
The warped solution of Einstein's equations corresponding to the spherical brane in five-dimensional AdS is considered. This metric represents interiors of black holes on both sides of the brane and can provide gravitational trapping of physical fields on the shell. It is found the analytic form of the coordinate transformations from the Schwartschild to co-moving frame that exists only in five dimensions. It is shown that in the static coordinates active gravitational mass of the spherical brane, in agreement with Tolman's formula, is negative, i.e. such objects are gravitationally repulsive.
We investigate the massive graviton contributions to 4D gravity in a 6D brane world scenario, whose bulk field content can include that of 6D chiral gauged supergravity. We consider a general class of solutions having 3-branes, 4D Poincare symmetry and axisymmetry in the internal space. We show that these contributions, which we compute analytically, can be independent of the brane vacuum energy as a consequence of geometrical and topological properties of the above-mentioned codimension two brane world. These results support the idea that in such models the gravitational interactions may be d…
A new solution to Einstein equations in (1+5)-spacetime with an embedded (1+3) brane is given. This solution localizes the zero modes of all kinds of matter fields and 4-gravity on the (1+3) brane by an increasing, transverse gravitational potential. This localization occurs despite the fact that the gravitational potential is not a decreasing exponential, and asymptotically approaches a finite value rather than zero.
In this paper, we mainly investigate the problems of resonances of massive KK fermions on a single scalar constructed thick brane with a piecewise warp factor matching smoothly. The distance between two boundaries and the other parameters are determined by one free parameter through three junction conditions. For the generalized Yukawa coupling $η\barΨϕ^{k}Ψ$ with odd $k=1,3,5,...$, the mass eigenvalue $m$, width $Γ$, lifetime $τ$, and maximal probability $P_{max}$ of fermion resonances are obtained. Our numerical calculations show that the brane without internal structure also favors the appe…
We present a higher dimensional model where gravity is bound to a brane due to Anderson localization. The extra dimensions are taken to be a disordered crystal of branes, with randomly distributed tensions of order the fundamental scale. Such geometries bind the graviton and thus allow for arbitrarily large extra dimensions even when the curvature is small. Thus this model is quite distinct from that of Randall and Sundrum where localization is a consequence of curvature effects in the bulk. The hierarchy problem can be solved by having the standard model brane live a distance away from the br…
We study the braneworld effective action in the two-brane Randall-Sundrum model. In the framework of this essentially-nonlocal action we reveal the origin of an infinite sequence of gravitational wave modes -- the usual massless one as well as the tower of Kaluza-Klein massive ones. Mixing of the modes, which parametrically depends on the background value of the modulus of the extra dimension, can be interpreted as radion-induced gravitational-wave oscillations, a classical analogue to meson and neutrino oscillations. We show that these oscillations arising in M-theory-inspired braneworld setu…
It is argued that D=10 type II strings and M-theory in D=11 have D-5 branes and 9-branes that are not standard p-branes coupled to anti-symmetric tensors. The global charges in a D-dimensional theory of gravity consist of a momentum $P_M$ and a dual D-5 form charge $K_{M_1...M_{D-5}}$, which is related to the NUT charge. On dimensional reduction, P gives the electric charge and K the magnetic charge of the graviphoton. The charge K is constructed and shown to occur in the superalgebra and BPS bounds in $D\ge 5$, and leads to a NUT-charge modification of the BPS bound in D=4. $K$ is carried by …
We present a new (1+3)-brane solution to Einstein equations in (1+5)-space. As distinct from previous models this solution is free of singularities in the full 6-dimensional space-time. The gravitational potential transverse to the brane is an increasing (but not exponentially) function and asymptotically approaches a finite value. The solution localizes the zero modes of all kinds of matter fields and Newtonian gravity on the brane. An essential feature of the model is that different kind of matter fields have different localization radii.
The effective gravitational field equations on and off a 3-brane world possessing a Z_{2} mirror symmetry and embedded in a five-dimensional bulk spacetime with cosmological constant were derived by Shiromizu, Maeda and Sasaki (SMS) in the framework of the Gauss-Codazzi projective approach with the subsequent specialization to the Gaussian normal coordinates in the neighborhood of the brane. However, the Gaussian normal coordinates imply a very special slicing of spacetime and clearly, the consistent analysis of the brane dynamics would benefit from complete freedom in the slicing of spacetime…
Plane wave solutions of Schrodinger-like equations obtained from the metric perturbations in 5D braneworld scenarios can present resonant modes. The search for those structures is important because they can provide us massive modes with not suppressed couplings with the membrane. We propose in this paper the study of graviton Kaluza-Klein spectrum in a special kind of membrane that possesses internal structure. The interest in study of these deformed defects is because they have a more rich internal structure that has implications in the matter-energy density along the extra dimensions an this…
Braneworld models are interesting theoretical and phenomenological frameworks to search for new physics beyond the standard model of particles and cosmology. In this work, we discuss braneworld models whose gravitational dynamics are governed by teleparallel $f(T)$ gravities. Here, we emphasize a codimension two axisymmetric model, also known as a string-like brane. Likewise, in the 5D domain-walls models, the $f(T)$ gravitational modification leads to a phase transition on the perfect fluid source providing a brane-splitting mechanism. Furthermore, the torsion changes the gravitational pertur…
In this paper, we examine the possibility of static, spherically symmetric gravitational geons on a 3 dimensional brane embedded in a 4+1 dimensional space-time. We choose a specific g_tt for the brane-world space-time metric. We then calculate g_rr analytically in the weak field limit and numerically for stronger fields. We show that the induced field equations on the brane do admit gravitational geon solutions.
In this paper, we will analyze the effects of super-light brane world perturbative modes on clustering of galaxies. In present manuscript, we use the Boltzmann and Tsallis statistical approaches to study the large distance modification of gravity in the brane world. The impact of modified potential on clustering of galaxies is analyzed in both the approaches. The infinities associated with Newtonian point mass approximation of gravity models is removed using the analytical extensions. The regularized and finite partition function is obtained for the system of galaxies in the brane world model …
The causal structure of the flat brane universe of RSII type is re-investigated to clarify the boundary conditions for stochastic gravitational waves. In terms of the Gaussian normal coordinate of the brane, a singularity of the equation for gravitational waves appears in the bulk. We show that this singularity corresponds to the ``seam singularity'' which is a singular subspace on the brane universe. Based upon the causal structure, we discuss the boundary conditions for gravitational waves in the bulk. Introducing a null coordinate, we propose a numerical procedure to solve gravitational wav…
We derive the full projected Einstein-Brans-Dicke gravitational equations associated with a n-dimensional brane embedded in a (n+1)-dimensional bulk. By making use of general conditions, as the positivity of the Brans-Dicke parameter and the effective Newton gravitational constant as well, we are able to constrain the brane cosmological constant in terms of the brane tension, the Brans-Dicke scalar field, and the trace of the stress tensor on the brane, in order to achieve a $dS$ brane. Applying these constraints to a specific five-dimensional model, a lower bound for the scalar field on the b…
In this paper we consider a static domain wall inside a 3-brane. Differently of the standard achievement obtained in General Relativity, the analysis performed here gives a consistency condition for the existence of static domain walls in a braneworld gravitational scenario. It is also shown the behavior of the domain wall gravitational field in the newtonian limit.
We first obtain the analogue of Vaidya's solution on the brane for studying the collapse of null fluid onto a flat Minkowski cavity on the brane. Since the back-reaction of the bulk onto the brane is supposed to strengthen gravity on the brane, it would favour formation of black hole as against naked singularity. That is the parameter window in the initial data set giving rise to naked singularity in the 4D Vaidya case would now get partially covered.
In a toy model we derive the gravitational equation on a self-gravitating curved D-brane. The effective theory on the brane is drastically changed from the ordinal Einstein equation. The net cosmological constant on the brane depends on a tuning between the brane tension and the brane charges. Moreover, non-zero matter stress tensor exists if the net cosmological constant is not zero. This fact indicates a direct connection between matters on the brane and the dark energy.
In this paper, we analyze the deflection angle of light by Brane-Dicke wormhole in the weak field limit approximation to find the effect of the Brane-Dicke coupling parameter on the weak gravitation lensing. For this purpose, we consider new geometric techniques, i.e., Gauss-Bonnet theorem and optical geometry in order to calculate the deflection angle. Furthermore, we verify our results by considering the most familiar geodesic technique. Moreover, we establish the quantum corrected metric of Brane-Dicke wormhole by replacing the classical geodesic with Bohmian trajectories, whose matter sour…
Following Campos's work [Phys. Rev. Lett. 88, 141602 (2002)], we investigate the effects of temperature on flat, de Sitter (dS), and anti-de Following Campos's work [Phys. Rev. Lett. \textbf{88}, 141602 (2002)], we investigate the effects of temperature on flat, de Sitter (dS), and anti-de Sitter (AdS) thick branes in five-dimensional (5D) warped spacetime, and on the fermion (quasi-)localization. First, in the case of flat brane, when the critical temperature reaches, the solution of the background scalar field and the warp factor is not unique. So the thickness of the flat thick brane is unc…
In this article we study a generalization of DGP scenarios, where the induced gravity is given by a $f(R)$ term. We obtain the effective gravitational equations and the effective FLRW cosmological equation on the brane of this model. We show that this generalization has also two regime, a 5D regime a low energies that has a self-accelerated branch of interest for cosmology and a 4D regime at high energies that it is described a modified gravitational theory.
In this work we investigate the issue of fermion localization and resonances in $(4,1)$-deformed branes constructed with one scalar field coupled with gravity. Such models provide us branes with internal structures that turns the gravitational interaction more effective for fermions aside the brane, increasing their lifetime. The coupling between the scalar field and spinors is a necessary condition for fermions to be localized on such branes. After performing a chiral decomposition of the five-dimensional spinor we found resonances with both chiralities. The correspondence between the spectra…
In this article, we develop a formalism which is different from the standard lensing scenario and is necessary for understanding lensing by gravitational fields which arise as solutions of the effective Einstein equations on the brane. We obtain general expressions for measurable quantities such as time delay, deflection angle, Einstein ring and magnification. Subsequently, we estimate the deviations (relative to the standard lensing scenario) in the abovementioned quantities by considering the line elements for clusters and spiral galaxies obtained by solving the effective Einstein equations …
We analyze the general covariant energy-momentum tensor of the gravitational system in genreal five-dimensional cosmological brane-world models. Then through calculating this energy-momentum for the cosmological generalization of the Randall-Sundrum model, which includes the original RS model as the static limit, we are able to show that the weakness of the gravitation on the "visible" brane is a general feature of this model. This is the origin of the gauge hierarchy from a gravitational point of view. Our results are also consistent with the fact that a gravitational system has vanishing tot…
A few years ago, Campos investigated the critical phenomena of thick branes in warped spacetimes [Phys. Rev. Lett. {\bf 88} (2002) 141602]. Inspired by his work, we consider a toy model of thick branes generated by a real scalar field with the potential $V(ϕ)=aϕ^2-bϕ^4+cϕ^6$, and investigate the variation of the mass parameter $a$ on the branes as well as the localization and resonances of fermions. An interesting result is found: there is a critical value for the mass parameter $a$, and when the critical value of $a$ is reached the solution of the background scalar field is not unique and has…
We calculate the amplitude of gravitational waves produced by inflation on a de Sitter brane embedded in five-dimensional anti-de Sitter bulk spacetime, extending previous calculations in Randall-Sundrum type cosmology to include the effect of induced gravity corrections on the brane. These corrections arise via a term in the brane action that is proportional to the brane Ricci scalar. We find that, as in the Randall-Sundrum case, there is a mass gap between the discrete zero-mode and a continuum of massive bulk modes, which are too heavy to be excited during inflation. We give the normalizati…
We derive the deflection angle of light rays caused by a brane black hole with mass m and tidal charge q in the weak lensing approach, up to the second order in perturbation theory. We point out when the newly derived second order contributions become important.
In brane-world cosmology gravitational waves can propagate in the higher dimensions (i.e., in the `bulk'). In some appropriate regimes the bulk gravitational waves may be approximated by plane waves. We systematically study five-dimensional gravitational waves that are algebraically special and of type N. In the most physically relevant case the projected non-local stress tensor on the brane is formally equivalent to the energy-momentum tensor of a null fluid. Some exact solutions are studied to illustrate the features of these branes; in particular, we show explicity that any plane wave brane…
We study the evolution of gravitational waves(GWs) after inflation in a brane-world cosmology embedded in five-dimensional anti-de Sitter spacetime. Contrary to the standard four-dimensional results, the GWs at the high-energy regime in brane-world model suffer from the effects of the non-standard cosmological expansion and the excitation of the Kaluza-Klein modes(KK-modes), which can affect the amplitude of stochastic gravitational wave background significantly. To investigate these two high-energy effects quantitatively, we numerically solve the wave equation of the GWs in the radiation domi…
The aim of this contribution is to provide a short introduction to recently investigated models in which our accessible universe is a four-dimensional submanifold, or brane, embedded in a higher dimensional spacetime and ordinary matter is trapped in the brane. I focus here on the gravitational and cosmological aspects of such models with a single extra-dimension.
