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Recent developments in string theory suggest that there might exist extra spatial dimensions, which are not small nor compact. The framework of most brane cosmological models is that the matter fields are confined on a brane-world embedded in five dimensions (the bulk). Motivated by this we re-examine the classification of the second-order symmetric tensors in 5-D, and prove two theorems which collect together some basic results on the algebraic structure of these tensors in five-dimensional spacetimes. We also briefly indicate how one can obtain, by induction, the classification of symmetric two-tensors (and the corresponding canonical forms) on n-dimensional (n>4) spaces from the classification on four-dimensional spaces. This is important in the context of 11-D supergravity and 10-D superstrings.

We propose a multi-graviton theory with non-nearest-neighbor couplings in the theory space. The resulting four-dimensional discrete mass spectrum reflects the structure of a latticed extra dimension. For a plausible mass spectrum motivated by the discretized Randall–Sundrum brane-world, the induced cosmological constant turns out to be positive and may serve as a quite simple model for the dark energy of our accelerating universe.

Recently it was suggested that the dark energy maybe related to the well-known hierarchy between the Planck scale (~ 10¹⁹GeV) and the TeV scale. The same brane-world setup to address this hierarchy problem may also in principle address the smallness problem of dark energy. Specifically, the Planck-SM hierarchy ratio was viewed as a quantum gravity-related, dimensionless fine structure constant where various physical energy scales in the system are associated with the Planck mass through different powers of the 'gravity fine structure constant'. In this paper we provide a toy model based on the Randall-Sundrum geometry where SUSY-breaking is induced by the coupling between a SUSY-breaking Higgs field on the brane and the KK gravitinos. We show that the associated Casimir energy density indeed conforms with the dark energy scale.

The topological structure of Schwazschild's space-time and its maximal analytic extension are investigated in the context of brane worlds. Using the embedding coordinates, these geometries are seen as different states of the evolution of a single brane-world. Comparing the topologies and the embeddings it is shown that this evolution must be followed by a signature change in the bulk.

In the Randall–Sundrum (RS) brane-world model a singular delta-function source is matched by the second derivative of the warp factor. So one should take possible curvature corrections in the effective action of the RS models in the Gauss–Bonnet (GB) form. We present a linearized treatment of gravity in the RS brane-world with the Gauss–Bonnet modifications to Einstein gravity. We give explicit expressions for the Neumann propagator in arbitrary D dimensions and show that a bulk GB term gives, along with a tower of Kaluza–Klein modes in the bulk, a massless graviton on the brane, as in the standard RS model. Moreover, a nontrivial GB coupling can allow a new branch of solutions with finite Planck scale and no naked bulk singularity, which might be useful to avoid some of the previously known "no-go theorems" for RS brane-world compactifications.

In the light of the Planck 2015 results, we update simple inflationary models based on the quadratic, quartic, Higgs and Coleman–Weinberg potentials in the context of the Randall–Sundrum brane-world cosmology. Brane-world cosmological effect alters the inflationary predictions of the spectral index $(n_s)$ and the tensor-to-scalar ratio $(r)$ from those obtained in the standard cosmology. In particular, the tensor-to-scalar ratio is enhanced in the presence of the 5th dimension. In order to maintain the consistency with the Planck 2015 results for the inflationary predictions in the standard cosmology, we find a lower bound on the five-dimensional Planck mass $(M_5)$. On the other hand, the inflationary predictions laying outside of the Planck allowed region can be pushed into the allowed region by the brane-world cosmological effect with a suitable choice of $M_5$.

Extra-dimensional theories contain additional degrees of freedom related to the geometry of the extra space which can be interpreted as new particles. Such theories allow to reformulate most of the fundamental problems of physics from a completely different point of view. In this essay, we concentrate on the brane fluctuations which are present in brane-worlds, and how such oscillations of the own space–time geometry along curved extra dimensions can help to resolve the Universe missing mass problem. The energy scales involved in these models are low compared to the Planck scale, and this means that some of the brane fluctuations distinctive signals could be detected in future colliders and in direct or indirect dark matter searches.

It is explained why the curvature associated with the vacuum energy density arising from SUSY breaking cannot be completely transferred to the extra spatial dimensions of a bulk space–time manifold, and it is shown — without using hierarchy arguments but only the results of current large-scale observations — why the TeV scale should correspond to the maximal allowed SUSY-breaking scale.

Higher-dimensional black holes have long been considered within the context of brane worlds. Recently, it was shown that the brane-world ethos also permits the consideration of higher-dimensional wormholes. When such a wormhole, pre-existing in the bulk, impinges upon our universe, taken to be a positive-tension three-brane, it can induce the creation in our universe of a wormhole of ordinary dimensionality. The throat of this wormhole might fully constrict, pinch off, and thus birth a baby universe. Alternatively, the induced wormhole might persist. I show that persistence is more likely and note that the persistent wormhole manifests itself as a particle-like object whose interaction with cosmic matter is purely gravitational. I consider the viability of this object as a dark matter candidate.

Brane-world models, where observers are trapped within the thickness of a 3-brane, offer novel perspectives on gravitation and cosmology. In this essay, it is argued that the problem of a late epoch acceleration of the universe is well explained in the framework of a 4-dimensional de Sitter universe embedded in a 5-dimensional de Sitter spacetime. While a 5-dimensional Anti-de Sitter space background is important for studying conformal field theories for its role in the AdS/CFT correspondence, the existence of a 5-dimensional de Sitter space is crucial for finding an effective 4-dimensional Newton constant that remains finite and a normalizable zero-mode graviton wave function.

In this paper, we study a constant-roll inflationary model in the context of brane-world cosmology caused by a quintessence scalar field for warm inflation with a constant dissipative parameter $Q=\mathrm{\Gamma }/3H$. We determine the analytical solution for the Friedmann equation coupled to the equation of motion of the scalar field. The evolution of the primordial scalar and tensor perturbations is also studied using the modified Langevin equation. To check the viability of the model, we use numerical approaches and plot some figures. Our results for the scalar spectral index and the tensor-to-scalar ratio show good consistency with observations.

The term dark radiation is used both to describe a noninteracting neutrino species and as a correction to the Friedmann Equation in the simplest five-dimensional (5D) RS-II brane-world cosmology. In this paper, we consider the constraints on both the meanings of dark radiation-based upon the newest results for light-element nuclear reaction rates, observed light-element abundances and the power spectrum of the Cosmic Microwave Background (CMB). Adding dark radiation during big bang nucleosynthesis (BBN) alters the Friedmann expansion rate causing the nuclear reactions to freeze out at a different temperature. This changes the final light element abundances at the end of BBN. Its influence on the CMB is to change the effective expansion rate at the surface of the last scattering. We find that the BBN constraint reduces the allowed range for both types of dark radiation at 10Mev to between $-12.1\%$ and $+6.2\%$ of the total background energy density at 10Mev. Combining this result with fits to the CMB power spectrum, produces different results for particle versus brane-world dark radiation. In the brane-world, the range decreases from $+6.2\%$ to $-6.0\%$. Thus, we find that the ratio of dark radiation to the background total relativistic mass energy density ${\rho }_{\mathrm{\text{DR}}}/\rho$ is consistent with zero although there remains a very slight preference for a positive (rather than negative) contribution.

Through the characterization of a spherically symmetric space-time as a local brane-world immersed into six-dimensional pseudo-Euclidean spaces, with different signatures of the bulk, we investigate the existence of a topological difference in the immersed brane-world. In particular the Schwarzschild's brane-world and its Kruskal (or Frønsdal) brane-world extension are examined from point of view of the immersion formalism. We prove that there is a change of signature of the bulk when we consider a local isometric immersion and different topologies of a brane-world in that bulk.

A brane-world universe consists of a 4-dimensional brane embedded into a 5-dimensional space-time (bulk). We apply the Arnowitt-Deser-Misner decomposition to the brane-world, which results in a 3+1+1 break-up of the bulk. We present the canonical theory of brane cosmology based on this decomposition. The Hamiltonian equations allow for the study of any physical phenomena in brane gravity. This method gives new prospects for studying the initial value problem, stability analysis, brane black holes, cosmological perturbation theory and canonical quantization in brane-worlds.

We give the evolutions and constraints obtained as differences of the higher dimensional dynamical equations across the asymmetrically embedded brane.

I present recent work on codimension one brane-world models containing a 3+1 dimensional curved brane with time-dependent brane tension.

The ADM canonical formulation of the gravitational field is extended to four-dimensional space-times embedded in a higher-dimensional bulk space. The embedding is justified as a means to remove the ambiguity of the Riemann curvature. Using Nash’s perturbative embedding theorem, we derive a simple generalization of the ADM canonical structure without breaking the diffeomorphism invariance and with a non-vanishing Hamiltonian. Nash’s perturbations also allows us to define the functional derivative in Schwinger’s equation, leading to a Schrödinger-like quantum equation describing the wave function of the embedded space-time.