Higher dimensional cosmologies

Einstein equations are derived for D-dimensional space-time that spontaneously compactify to the product M⁴ × Π_{i = 1}^α M^d_i in which the metric is taken to be of the generalized Robertson-Walker form. Cosmological solutions for these equations are studied with power law, oscillatory and exponential behaviour for the D-dimensional Einstein-Maxwell, N = 2, D = 10 and N = 1, D = 11 supergravity models. In the Einstein-Maxwell case the presence of a cosmological constant forces the extra dimensions to be static. Nevertheless, it is possible to find solutions with vanishing effective 4 dimensional cosmological constant with an expanding 4-dimensional space-time. In the supergravity models the requirement of having compact extra dimensions restricts the solutions to have expansion only in the 4-dimensional space-time. Matter contribution is added to the energy-momentum tensor in an attempt to find new solutions.     

Unbounded Entropy in Spacetimes with Positive Cosmological Constant

In theories of gravity with a positive cosmological constant, we consider product solutions with flux, of the form (A)dS _p ×S ^q . Most solutions are shown to be perturbatively unstable, including all uncharged dS _p ×S ^q spacetimes. For dimensions greater than four, the stable class includes universes whose entropy exceeds that of de Sitter space, in violation of the conjectured N-bound. Hence, if quantum gravity theories with finite-dimensional Hilbert space exist, the specification of a positive cosmological constant will not suffice to characterize the class of spacetimes they describe.     

Primordial inflation and present-day cosmological constant from extra dimensions

A semiclassical gravitation model is outlined which makes use of the Casimir energy density of vacuum fluctuations in extra compactified dimensions to produce the present-day cosmological constant as ρ _Λ ∼M ⁸/M _P ⁴, where M _P is the Planck scale and M is the weak interaction scale. The model is based on (4+D)-dimensional gravity, with D=2 extra dimensions with radius b(t) curled up at the ADD length scale b ₀=M _P /M ²∼0.1 mm. Vacuum fluctuations in the compactified space perturb b ₀ very slightly, generating a small present-day cosmological constant.The radius of the compactified dimensions is predicted to be b ₀≈k ^1/40.09 mm (or equivalently M≈2.4 TeV/k ^1/8), where the Casimir energy density is k/b ⁴.Primordial inflation of our three-dimensional space occurs as in the cosmology of the ADD model as the inflaton b(t), which initially is on the order of 1/M∼10⁻¹⁷ cm, rolls down its potential to b ₀.     

Charge and/or spin limits for black holes at a non-commutative scale

We study the cosmological dynamics for R ^p exp(λ R) gravity theory in the metric formalism, using dynamical systems approach. Considering higher-dimensional FRW geometries in case of an imperfect fluid which has two different scale factors in the normal and extra dimensions, we find the exact solutions, and study its behaviour and stability for both vacuum and matter cases. It is found that stable solutions corresponding to accelerated expansion at late times exist, which can describe the inflationary era of the Universe. We also study the evolution of scale factors both in the normal and extra dimensions for different values of anisotropy parameter and the number of extra dimensions for such a scenario.     

On the initial conditions for super-exponential inflation

In a higher-dimensional theory of gravity containing higher-derivative terms and a cosmological constant, a period of super-exponential inflation of the physical spacetime is possible, during which the Hubble parameter H increases with time, as discovered by Shafi and Wetterich. This means that the initial value H₀ must be small H₀ < 6 × 10⁻⁵m_p. We discuss the meaning of this initial condition from the standpoint of quantum cosmology.     

Modification of the Coulomb Potential from a Kaluza-Klein Model with a Gauss-Bonnet Term in the Action

In four dimensions a Gauss-Bonnet term in the action corresponds to a total derivative, and therefore it does not contribute to the classical equations of motion. For higher-dimensional geometries this term has the interesting property (which it shares with other dimensionally continued Euler densities) that when the action is varied with respect to the metric, it gives rise to a symmetric, covariantly conserved tenser of rank two which is a function of the metric and its first- and second-order derivatives. Here we review the unification of general relativity and electromagnetism in the classical five-dimensional, restricted (with g₅₅ = 1) Kaluza-Klein model. Then we discuss the modifications of the Einstein-Maxwell theory that results from adding the Gauss-Bonnet term in the action. The resulting four-dimensional theory describes a non-linear U(1) gauge theory non-minimally coupled to gravity. For a point charge at rest we find a perturbative solution for large distances which gives a mass-dependent correction to the Coulomb potential. Near the source we find a power-law solution which seems to cure the short-distance divergency of the Coulomb potential. Possible ways to obtain an experimental upper limit to the coupling of the hypothetical Gauss-Bonnet term are also considered.     

Flux vacua in DBI type Einstein–Maxwell theory

We study compactification of extra dimensions in a theory of Dirac–Born–Infeld type gravity. We investigate the solution for Minkowski spacetime with an S ² extra space as well as that for de Sitter spacetime (S ⁴) with an S ² extra space. They are derived by the effective potential method in the presence of the magnetic flux on the extra sphere. We also consider the higher-dimensional generalization of the solutions. We find that, in a certain model, the radius of the extra space has a minimum value independent of the higher-dimensional Newton constant.     

Warped de Sitter compactifications in the scalar–tensor theory

We present new solutions of warped compactifications in the higher-dimensional gravity coupled to the scalar and the form field strengths. These solutions are constructed in the D-dimensional spacetime with matter fields, with the internal space that has a finite volume. Our solutions give explicit examples where the cosmological constant or 0-form field strength leads to a de Sitter spacetime in arbitrary dimensions.     

Screening of cosmological constant for de Sitter Universe in non-local gravity,phantom-divide crossing and finite-time future singularities

We investigate de Sitter solutions in non-local gravity as well as in non-local gravity with Lagrange constraint multiplier. We examine a condition to avoid a ghost and discuss a screening scenario for a cosmological constant in de Sitter solutions. Furthermore, we explicitly demonstrate that three types of the finite-time future singularities can occur in non-local gravity and explore their properties. In addition, we evaluate the effective equation of state for the universe and show that the late-time accelerating universe may be effectively the quintessence, cosmological constant or phantom-like phases. In particular, it is found that there is a case in which a crossing of the phantom divide from the non-phantom (quintessence) phase to the phantom one can be realized when a finite-time future singularity occurs. Moreover, it is demonstrated that the addition of an R ² term can cure the finite-time future singularities in non-local gravity. It is also suggested that in the framework of non-local gravity, adding an R ² term leads to possible unification of the early-time inflation with the late-time cosmic acceleration.     

Limits on low scale gravity from e+e−→W+W−, ZZ and γγ

It has been proposed recently that the scale of quantum gravity (“the string scale”) can be M_S∼few TeV with n≥2 extra dimensions of size R≲mm so that, at distances greater than R, Newtonian gravity with M_Pl∼10¹⁸ GeV is reproduced if M_Pl²∼RⁿM_Sⁿ⁺². Exchange of virtual gravitons in this theory generates higher-dimensional operators involving SM fields, suppressed by powers of M_S. We discuss constraints on this scenario from the contribution of these operators to the processes e⁺e⁻→W⁺W⁻, ZZ, γγ. We find that LEP2 can place a limit M_S≈1 TeV from e⁺e⁻→W⁺W⁻, ZZ, γγ.     

Higher-dimensional cosmological solutions with action of scalar and metric fields

An investigation is made of higher-dimensional (D5) cosmological solutions with action of scalar and metric fields for which a matter term is added. We restrict our attention to the most symmetric solutions with the structureM ^D–2×S ². We present the variant cosmological solutions for the symmetry breaking patternGSU(2)×U(1) (type IA, IIA) and patternGSO(3) (type IB, IIB). InD=6 case type IA is interesting for cosmology, which corresponds to a conformally invariant theory.     

Cosmological application on five-dimensional teleparallel theory equivalent to general relativity

A theory of(4+1)-dimensional gravity has been developed on the basis of which equivalent to the theory of general relativity by teleparallel.The fundamental gravitational field variables are the 5-dimensional(5D) vector fields(pentad),defined globally on a manifold M,and gravity is attributed to the torsion.The Lagrangian density is quadratic in the torsion tensor.We then apply the field equations to two different homogenous and isotropic geometric structures which give the same line element,i.e.,FRW in five dimensions.The cosmological parameters are calculated and some cosmological problems are discussed.     

Cosmological constant: a lesson from the effective gravity of topological weyl media

Topological matter with Weyl points, such as superfluid ³He-A, provide an explicit example where there is a direct connection between the properly determined vacuum energy and the cosmological constant of the effective gravity emerging in condensed matter. This is in contrast to the acoustic gravity emerging in Bose-Einstein condensates (S. Finazzi, S. Liberati, and L. Sindoni, Phys. Rev. Lett. 108, 071101 (2012); arXiv:1103.4841). The advantage of topological matter is that the relativistic fermions and gauge bosons emerging near the Weyl point obey the same effective metric and thus the effective gravity is more closely related to real gravity. We study this connection in the bi-metric gravity emerging in ³He-A, and its relation to the graviton masses, by comparison with a fully relativistic bi-metric theory of gravity. This shows that the parameter ??, which in ³He-A is the bi-metric generalization of the cosmological constant, coincides with the difference in the proper energy of the vacuum in two states (the nonequilibrium state without gravity and the equilibrium state in which gravity emerges) and is on the order of the characteristic Planck energy scale of the system. Although the cosmological constant ?? is huge, the cosmological term T _??? ^?? itself is naturally non-constant and vanishes in the equilibrium vacuum, as dictated by thermodynamics. This suggests that the equilibrium state of any system including the final state of the Universe is not gravitating.     

“Regge-like” relations for stable (non-evaporating) black holes

Within a purely classical formulation of “strong gravity,” we associated hadron constituents (and even hadrons themselves) with suitable stationary, axisymmetric solutions of certain new Einsteintype equations supposed to describe the strong field inside hadrons. Such equations are nothing but Einstein equations—with cosmological term—suitably scaled down. As a consequence, the cosmological constant Λ and the massesM result in our theory to be scaled up, and transformed into a “hadronic constant” and into “strong masses,” respectively. Due to the unusual range of Λ andM values considered, we met a series of solutions of the Kerr-Newman-de Sitter (KNdS) type with rather interesting properties: aim of the present work is putting forth such results, while “translating” them into the more popular language of ordinary gravity. The requirement that those solutions be stable, i.e., that their temperature (or surface gravity) bevanishingly small, implies the coincidence of at least two of their (in general, three) horizons. Imposing the stability condition of a certain horizon does yield (once chosen the values ofJ, q and Λ) mass and radius of the associated black hole. In the case of ordinary Einstein equations and for stable blackholes of the KNdS type, we get in particular Regge-like relations among massM, angular momentumJ, chargeq and cosmological constant Λ; which did not receive enough attention in the previous literature. For instance, with the standard definitionsQ ² = Gq²/(4πε ₀ c ⁴), a ≡ J/(Mc), m ≡GM/c ², in the case Λ=0 in whichm ²=a²+Q² and ifq is negligible, we findm ²=J. When considering, for simplicity, Λ>0 andJ=0 (andq still negligible), then we obtainm ² = 1/(9Λ). In the most general case, the condition, for instance, of “triple coincidence” among the three horizons yields for |Λa ²|<< 1 the couple of independent relationsm ² = 2/(9Λ) andm ² = 8(a ² + Q². Another interesting point is that—with few exceptions—all such relations (amongM, J, q, Λ) lead to solutions that can be regarded as (stable) cosmological models. Work partially supported by INFN, MURST, and CNR and by CNPq, FAPESP, and CAPES.     

Einstein's gravity from a first order lagrangian in an embedding space

We formulate a first order action principle in a higher dimensional space M_N in which we embed spacetime. The action I is essentially an “area” of a four-dimensional spacetime V₄ weighted with a matter density ω in M_N. For a suitably chosen ω we obtain on V₄ a set of worldlines. It is shown that these worldlines are geodesics of V₄, provided that V₄ is a solution to our variational procedure. Then it follows that our spacetime satisfies the Einstein equations for dust - apart from an additional term with zero covariant divergence. (This extra term was shown in a previous paper to be exactly zero at least in the case of the cosmological dust model.) Thus we establish a remarkable connection of the extrinsic spacetime theory with the intrinsic general relativity. This step appears to be important for quantum gravity.     

(1+2)-Dimensional model of the early universe

An anisotropic cosmological model is obtained by solving (1+3)-dimensional field equations. The topology of the model isR ¹ M ² S ¹, whereR ¹ is the real line (time axis),M ² is 2-dimensional space, andS ¹ is the circle. Employing the method of Kaluza-Klein type compactification onS ¹ and one-loop quantum correction to scalar fields, an effective (1+2)-dimensional gravity is obtained. The resulting (1+2)-dimensional cosmological model of the early universe is derived.     

Cosmological Density Perturbations from a Quantum Gravitational Model of Inflation

We derive the implications for anisotropies in the cosmic microwave background following from a model of inflation in which a bare cosmological constant is gradually screened by an infrared process in quantum gravity. The model predicts that the amplitude of scalar perturbations is A_S = (2.0 ± 0.2) · 10^—5, that the tensor-to-scalar ratio is r ≈︂ 1.7 · 10^—3, and that the scalar and tensor spectral indices are n ≈︂ 0.97 and n_T ≈︂ —2.8 · 10^—4, respectively. By comparing the model's power spectrum with the COBE 4-year RMS quadrupole, the mass scale of inflation is determined to be M = (0.72 ± 0.03) · 10¹⁶ GeV. At this scale the model produces about 10⁸ e-foldings of inflation, so another prediction is Ω = 1. PACS numbers: 04.60.-m, 98.80.Cq     

Einstein and Brans–Dicke Frames in Multidimensional Cosmology

Inhomogeneous multidimensional cosmological models with a higher-dimensional space-time manifold _{0 i=1} ⁿ M_i (n 1) are in stigated under dimensional reduction to a D ₀-dimensional effective non-minimally coupled -model which generalizes the familiar Brans–Dicke model. The general form of the Einstein frame representation of multidimensional solutions known in the Brans–Dicke frame is given with respect to cosmic synchronous time. As an example, the transformation is demonstrated explicitly for the generalized Kasner solutions where it is shown that solutions in the Einstein frame show no inflation of the external space although they can undergo deflation after the cosmic synchronous time inversion.     

Higher-dimensional Bianchi cosmologies

We present new higher-dimensional Bianchi cosmologies of class A. Our solutions given are of type Mⁿ = R × M³ × T^D where N = M³ are of types I, II, VI₀, VII⁰, VIII and IX. This spectrum of solutions includes the higher-dimensional versions of the Kasner solution and the mixmaster universe with stiff matter content.     

五维时空中度规的计算

在四维R+R2引力理论中给出了Wheeler-Dewitt(W-D)方程,通过分离变量法得到了W-D方程的解.利用Kaluza-Klein理论将Robertson-Walker度规推广到五维时空,结合时空中的场方程得到宇宙项与能量之间的关系.