A note on the generalized Friedmann equations for a thick brane

Within our thick brane approach previously used to obtain the cosmological evolution equations on a thick brane embedded in a five-dimensional Schwarzschild Anti-de Sitter spacetime it is explicitly shown that the consistency of these equations with the energy conservation equation requires that, in general, the thickness of the brane evolves in time. This varying brane thickness entails the possibility that both Newton’s gravitational constant G and the effective cosmological constant Λ₄ are time dependent.     

Brane cosmological evolution with bulk matter

We consider the cosmological evolution of a brane for general bulk matter content. In our setup the bulk pressure and the energy exchange densities are comparable to the brane energy density. Adopting a phenomenological fluid ansatz and generalizations of it, we derive a set of exact solutions of the Friedmann equation that exhibit accelerated expansion. We find that the effective equation of state parameter for the dark energy can exhibit w=−1$w=-1$ crossing without the presence of exotic matter.     

Spherically Symmetric Solutions and Dark Matter on the Brane

It has recently been suggested that our universe is a three-brane embedded in a higher dimensional spacetime. In this paper I examine static, spherically symmetric solutions that satisfy the effective Einstein field equations on a brane embedded in a five dimensional spacetime. The field equations involve a term depending on the five dimensional Weyl tensor, so that the solutions will not be Schwarzschild in general. This Weyl term is traceless so that any solution of ⁽⁴⁾ R = 0 is a possible four dimensional spacetime. Different solutions correspond to different five dimensional spacetimes and to different induced energy-momentum tensors on the brane. One interesting possibility is that the Weyl term could be responsible for the observed dark matter in the universe. It is shown that there are solutions of the equation ⁽⁴⁾ R = 0 that can account for the observed rotation curves of spiral galaxies.     

Brane Cosmology, Varying Speed of Light and Inflation in Models with One or More Extra Dimensions

We summarize the approach to brane cosmology known as mirage cosmology and use it to determine the Friedmann equation on a 3-brane embedded in different bulk spacetimes all with one or more extra dimensions. Usually, when there is more than one extra dimension the junction conditions, central to the usual brane world scenarios, are difficult to apply. This problem does not arise in mirage cosmology because the brane is treated as a test particle in the background spacetime. We discuss in detail the dynamics of a brane embedded in two specific 10D bulk spacetimes, namely Sch-AdS₅ × S₅ and a rotating black hole, and from the dynamics—which are now rather more complicated since the brane can move in all the extra dimensions—determine the new dark fluid terms in the brane Friedmann equation. Some of these, such as the cosmological constant term, are seen to be bulk dependent. We then show explicitly how this mirage cosmology approach matches with the familiar junction condition approach when there is just one extra dimension. The issue of a varying speed of light in mirage cosmology is addressed and we find a scenario in which c _eff always increases, tending asymptotically to c ₀ as the universe expands. Finally some comments are made regarding brane inflation and limitations of the mirage cosmology approach are also discussed.     

Evolution of Bouncing Cosmological Models Due to the Self-Gravitational Corrections

A four-dimensional timelike brane with non-zero energy density is considered as the boundary of a five dimensional Schwarzschild anti de Sitter bulk background. The self-gravitational corrections to the first Friedmann equation act as a source of stiff matter contrary to standard FRW cosmology where the charge of the black hole plays this role. In a previous related paper (Setare in Eur. Phys. J. C 47:851, [2006]), bouncing cosmology was studied, from a holographic perspective, for the very special case of a brane that is void of any intrinsic matter sources. In this paper we extend the results of (Setare in Eur. Phys. J. C 47:851, [2006]). We consider the physically relevant case in which a perfect fluid with equation of state of radiation is present on the brane. Then, we describe solutions of the braneworld theory under investigation and also determine their stability. Specifically, if we do not consider the self-gravitational corrections, the AdS black hole with zero ADM mass, and open horizon is an attractor, while, if we consider, the AdS black hole with zero ADM mass and flat horizon, and D3-brane with non-zero energy density is a repeller.     

Might Dark Matter and Energy be Intrinsic Properties of?Space?

It is shown that if a volume element V, of space is assumed to have intrinsic energy E, then basic principles of mechanics, thermodynamics and special relativity lead to the equation of state: E=pV, where p is the pressure. When this equation of state is incorporated in the Einstein equations it leads to the prediction that the orbital speed of matter circling a visible galaxy embedded in a spherical galactic halo should be relativistic, in disagreement with observations. However, it also leads directly to the interesting notion that the inertial mass of such a medium can be understood as a resistance to being compressed via Lorentz contraction. It is then shown that the mathematical structure of thermodynamics suggests another plausible definition of pressure if we allow for the possibility that the intrinsic energy of spacetime may not be described by the same work-energy relationship as ordinary matter. This additional possibility leads to the equation of state: E=−pV. While both of these equations of state describe forms of energy that are quite unlike ordinary energy, neither alone is able to account for observed rotational velocity curves of matter orbiting visible galaxies. Therefore, the possibility that space has two distinct components of energy is investigated. This results in a plausible, two-component equation of state in which the former equation of state is tentatively identified as the “dark matter” (DM) component, the latter as the “dark energy” (DE) component. The effective equation of state of space, accounting for the presence of both components, may then be written in the form: p=w ε, where ε is the total energy density, p the total pressure, and w represents the fractional excess of DM over DE (and therefore satisfies: −1≤w≤+1). Given the wide range of possible spacetime properties implied by this equation it appears to be a viable candidate for explaining observations presently attributed to the presence of both DM and DE. Specifically, the static, spherically symmetric solution of Einstein’s field equations, neglecting effects of ordinary matter, predicts the inverse r ² distribution of intrinsic space energy required to explain observed constant rotational velocity curves for matter in circular orbits around visible galaxies embedded within spherically symmetric galactic halos. The proposed equation of state is also capable of describing regions of space undergoing accelerated expansion as regions where DE is dominant (i.e., w<−1/3).     

Causality of the brane universe

Causal structure of the brane universe with respect to null geodesics in the bulk spacetime is studied. It is pointed out that apparent causality violation is possible for the brane universe which contains matter energy. It is also shown that there is no "horizon problem" in the Friedmann-Robertson-Walker brane universe.     

The Friedmann equation in brane-world scenarios

The Friedmann law on the brane generically depends quadratically on the brane energy density and involves a "dark radiation" term due to the bulk Weyl tensor. Despite its unfamiliar form, we show how it can be derived from a standard four-dimensional Brans-Dicke theory at low energy. In particular, the dark radiation term is found to depend linearly on the brane energy densities. For any equation of state on the branes, the radion evolves such as to generate radiation-dominated cosmology. The radiation-dominated era is conventional and consistent with nucleosynthesis.     

Induced cosmology on a regularized brane in six-dimensional flux compactification

We consider a six-dimensional Einstein–Maxwell system compactified in an axisymmetric two-dimensional space with one capped regularized conical brane of codimension one. We study the cosmological evolution which is induced on the regularized brane as it moves in between known static   bulk and cap solutions. Looking at the resulting Friedmann equation, we see that the brane cosmology at high energies is dominated by a five-dimensional ρ²${\rho }^2$ energy density term. At low energies, we obtain a Friedmann equation with a term linear to the energy density with, however, negative coefficient in the small four-brane radius limit (i.e., with negative effective Newton's constant). We discuss ways out of this problem.     

Time Variable Λ and the Accelerating Universe

We perform a deductive study of accelerating Universe and focus on the importance of variable time-dependent Λ in the Einstein’s field equations under the phenomenological assumption, Λ=αH ² for the full physical range of α. The relevance of variable Λ with regard to various key issues like dark matter, dark energy, geometry of the field, age of the Universe, deceleration parameter and barotropic equation of state has been trivially addressed. The deceleration parameter and the barotropic equation of state parameter obey a straight line relationship for a flat Universe described by Friedmann and Raychaudhuri equations. Both the parameters are found identical for α=1.     

Cosmological Kaluza–Klein branes in black brane spacetimes

We discuss the cosmological evolution of a brane in the D(>6)$D(>6)$-dimensional black brane spacetime in the context of the Kaluza–Klein (KK) braneworld scheme, i.e., to consider KK compactification on the brane. The bulk spacetime is composed of two copies of a patch of D  -dimensional black three-brane solution. The near-horizon geometry is given by AdS₅×S(D−5)${\mathit{AdS}}_5\times S^{(D-5)}$ while in the asymptotic infinity the spacetime approaches D-dimensional Minkowski. We consider the brane motion from the near-horizon region toward the spatial infinity, which induces cosmology on the brane. As is expected, in the early times, namely when the brane is located in the near-horizon region, the effective cosmology on the brane coincides with that in the second Randall–Sundrum (RS II) model. Then, the brane cosmology starts to deviate from the RS type one since the dynamics of KK compactified dimensions becomes significant. We find that the brane Universe cannot reach the asymptotic infinity, irrespectively of the components of matter on the brane.     

QCD Ghost Dark Energy in RS II Braneworld with Bulk-Brane Interaction

We investigate the QCD ghost model of dark energy in the framework of RS II braneworld. We assume there is an energy flow between the brane and bulk, and hence the continuity equation for the ghost dark energy is violated, while it is still preserved for the dark matter on the brane. We find that with the brane-bulk interaction, the equation of state parameter of ghost dark energy on the brane, can cross the phantom line w _D =?1 at the present time, which confirms by some cosmological evidences. This result is in contrast to the standard cosmology where w _D of ghost dark energy never cross the phantom line and the universe enters a de Sitter phase at the late time.     

Brane world cosmology without the Z2 symmetry

The Friedmann equation for a positive tension brane situated between two bulk spacetimes that posses the same 5D cosmological constant, but which does not posses a Z₂ symmetry of the metric itself is derived, and the possible effects of dropping the Z₂ symmetry on the expansion of our Universe are examined; cosmological constraints are discussed. We show the effect of this is an inflation-like period at very early times. The global solutions for the metric in the infinite extra dimension case are found and comparison with the symmetric case is made. We show that any brane world senario of this type must revert to a Z₂ symmetric form at late times, and hence rule out certain proposed scenarios.     

Cosmology of a brane radiating gravitons into the extra dimension

We study in a self-consistent way the impact of the emission of bulk gravitons on the (homogeneous) cosmology of a three-brane embedded in a five-dimensional spacetime. In the low energy regime, we recover the well known result that the bulk affects the Friedmann equation only via a radiationlike term C/a(4), called dark or Weyl radiation. In the high energy regime, we find that the Weyl parameter C is no longer constant but grows rapidly. Consequently, C today is determined by the past history of the brane universe and depends on the number of relativistic degrees of freedom at the high/low energy transition.     

Cutoff anti-de Sitter space/conformal-field-theory duality and the quest for braneworld black holes

Significant evidence is presented in favor of the holographic conjecture that "4D black holes localized on the brane found by solving the classical bulk equations in AdS5 are quantum corrected black holes and not classical ones." The quantum correction to the Newtonian potential is computed using a numerical computation of in Schwarzschild spacetime for matter fields in the zero-temperature Boulware vacuum state. For the conformally invariant scalar field the leading order term is equivalent to that previously obtained in the weak-field approximation using Feynman diagrams and which has been shown to be equivalent, via the anti-de Sitter space/conformal-field-theory (AdS/CFT) duality, to the analogous calculation in Randall-Sundrum braneworlds. The 4D backreaction equations are used to make a prediction about the existence and the possible spacetime structure of macroscopic static braneworld black holes.     

Eikonal equation of the Lorentz-violating Maxwell theory

We derive the eikonal equation of light wavefront in the presence of Lorentz invariance violation (LIV) from the photon sector of the standard model extension (SME). The results obtained from the equations of the E and B fields, respectively, are the same. This guarantees the self-consistency of our derivation. We adopt a simple case with only one non-zero LIV parameter as an illustration, from which we find two points. One is that, in analogy with the Hamilton–Jacobi equation, from the eikonal equation, we can derive dispersion relations which are compatible with results obtained from other approaches. The other is that the wavefront velocity is the same as the group velocity, as well as the energy flow velocity. If further we define the signal velocity v _s as the front velocity, there always exists a mode with v _s >1; hence causality is violated classically. Thus, our method might be useful in the analysis of Lorentz violation in QED in terms of classical causality.     

Modified gravity on the brane and dark energy

We analyze the dynamics of an AdS₅ braneworld with matter fields when gravity is allowed to deviate from the Einstein form on the brane. We consider exact five-dimensional warped solutions which are associated with conformal bulk fields of weight –4 and describe on the brane the following three dynamics: those of inhomogeneous dust, of generalized dark radiation, and of homogeneous polytropic dark energy. We show that, with modified gravity on the brane, the existence of such dynamical geometries requires the presence of non-conformal matter fields confined to the brane.     

5-dimensional braneworld with gravitating Nambu–Goto matching conditions

We continue the investigation of a recent proposal on alternative matching conditions for self-gravitating defects which generalize the standard matching conditions. The reasoning for this study is the need for consistency of the various codimension defects and the existence of a meaningful equation of motion at the probe limit, things that seem to lack from the standard approach. These matching conditions arise by varying the brane–bulk action with respect to the brane embedding fields (and not with respect to the bulk metric at the brane position) in a way that takes into account the gravitational back-reaction of the brane to the bulk. They always possess a Nambu–Goto probe limit and any codimension defect is seemingly consistent for any second order bulk gravity theory. Here, we consider in detail the case of a codimension-1 brane in five-dimensional Einstein gravity, derive the generic alternative junction conditions and find the Z₂$Z_2$-symmetric braneworld cosmology, as well as its bulk extension. Compared to the standard braneworld cosmology, the new one has an extra integration constant which accounts for the today matter and dark energy contents, therefore, there is more freedom for accommodating the observed cosmic features. One branch of the solution possesses the asymptotic linearized LFRW regime. We have constrained the parameters so that to have a recent passage from a long deceleration era to a small today acceleration epoch and we have computed the age of the universe, consistent with current data, and the time-varying dark energy equation of state. For a range of the parameters it is possible for the presented cosmology to provide a large acceleration in the high energy regime.     

Bulk Viscous Bianchi Type I Cosmological Models with Time-Dependent Cosmological Term Λ

Bianchi type I cosmological models with time-varying cosmological constant Λ and bulk viscous fluid are investigated. Cosmic matter is chosen to obey a barotropic equation of state. Exact solutions of Einstein’s field equations are obtained assuming the volume expansion θ proportional to the eigen values of shear tensor σ _ij . Physical and kinematical properties of the models are discussed considering bulk viscosity to be a power function of matter density.