Robertson Walker Space Research Articles

Density fluctuations in aκ-deformed inflationary universe

We study the spectrum of metric fluctuation in $\ensuremath{\kappa}$-deformed inflationary universe. We write the theory of scalar metric fluctuations in the $\ensuremath{\kappa}$-deformed Robertson-Walker space, which is represented as a nonlocal theory in the conventional Robertson-Walker space. One important consequence of the deformation is that the mode generation time is naturally determined by the structure of the $\ensuremath{\kappa}$-deformation. We expand the nonlocal action in ${H}^{2}/{\ensuremath{\kappa}}^{2}$, with $H$ being the Hubble parameter and $\ensuremath{\kappa}$ the deformation parameter, and then compute the power spectra of scalar metric fluctuations both for the cases of exponential and power law inflations up to the first order in ${H}^{2}/{\ensuremath{\kappa}}^{2}$. We show that the power spectra of the metric fluctuation have nontrivial corrections on the time dependence and on the momentum dependence compared to the commutative space results. Especially for the power law inflation case, the power spectrum for UV modes is weakly blueshifted early in the inflation and its strength decreases in time. The power spectrum of far-IR modes has cutoff proportional to ${k}^{3}$ which may explain the low cosmic microwave background (CMB) quadrupole moment.

Induced matter: Curved N-manifolds encapsulated in Riemann-flat N+1 dimensional space

Liko and Wesson have recently introduced a new five-dimensional induced matter solution of the Einstein equations, a negative curvature Robertson-Walker space embedded in a Riemann-flat five-dimensional manifold. We show that this solution is a special case of a more general theorem prescribing the structure of certain N+1 dimensional Riemann-flat spaces which are all solutions of the Einstein equations. These solutions encapsulate N-dimensional curved manifolds. Such spaces are said to “induce matter” in the submanifolds by virtue of their geometric structure alone. We prove that the N-manifold can be any maximally symmetric space.

NONCOMMUTATIVE SCALAR FIELD MINIMALLY COUPLED TO GRAVITY

A model for noncommutative scalar fields coupled to gravity based on the generalization of the Moyal product is proposed. Solutions compatible with homogeneous and isotropic flat Robertson-Walker spaces to first non-trivial order in the perturbation of the star-product are presented. It is shown that in the context of a typical chaotic inflationary scenario, at least in the slow-roll regime, noncommutativity yields no observable effect.

ENERGY–MOMENTUM IN VISCOUS KASNER-TYPE UNIVERSE IN BERGMANN–THOMSON FORMULATIONS

Using the Bergmann–Thomson energy–momentum complex and its tele-parallel gravity version, we obtain the energy and momentum of the universe in viscous Kasner-type cosmological models. The energy and momentum components (due to matter plus field) are found to be zero and this agree with a previous work of Rosen and Johri et al. who investigated the problem of the energy in Friedmann–Robertson–Walker universe. The result that the total energy and momentum components of the universe in these models is zero supports the viewpoint of Tryon. Rosen found that the energy of the Friedmann–Robertson–Walker space–time is zero, which agrees with the studies of Tryon.

Conformal symmetries of Einstein’s field equations and initial data

This paper examines the initial data for the evolution of the space–time solution of Einstein’s equations admitting a conformal symmetry. Under certain conditions on the extrinsic curvature of the initial complete spacelike hypersurface and sectional curvature of the space–time with respect to sections containing the normal vector field, we have shown that the initial hypersurface is conformally diffeomorphic to a sphere or a flat space or a hyperbolic space or the product of an open real interval and a complete 2-manifold. It has been further shown that if the initial hypersurface is compact, then it is conformally diffeomorphic to a sphere. Finally, the conformal symmetries of a generalized Robertson–Walker space–time have been described.

Some remarks on locally conformally flat static space–times

Necessary and sufficient conditions for a static space–time to be locally conformally flat are obtained, showing some significant restrictions on the possible warping functions of the space–times. This occurs in opposition to cosmological models, where Robertson–Walker space–times are locally conformally flat for any warping function.

Ideally embedded space–times

Due to the growing interest in embeddings of space–time in higher-dimensional spaces we consider a specific type of embedding. After proving an inequality between intrinsically defined curvature invariants and the squared mean curvature, we extend the notion of ideal embeddings from Riemannian geometry to the indefinite case. Ideal embeddings are such that the embedded manifold receives the least amount of tension from the surrounding space. Then it is shown that the de Sitter spaces, a Robertson–Walker space–time and some anisotropic perfect fluid metrics can be ideally embedded in a five-dimensional pseudo-Euclidean space.

Interacting Gravitational and Dirac Fields with Torsion

A general interaction scheme is formulated in a general space–time with torsion from the action principle by considering the gravitational, the Dirac, and the torsion field as independent fields. Some components of the torsion field come out to be automatically zero. Both the resulting Einstein-like and the Dirac-like fields equations contain nonlinear terms given by a self-interaction of the Dirac spinor and originally produced by torsion. The theory is specialized to the Robertson–Walker space–time without torsion. To solve he corresponding equations, that still have a complex structure, the spin coefficients have to be calculated explicitly from the tetrad employed. A solution, even if simple and elementary, is then determined.

Spacelike energy of timelike unit vector fields on a Lorentzian manifold

On a Lorentzian manifold, we define a new functional on the space of unit timelike vector fields given by the L2 norm of the restriction of the covariant derivative of the vector field to its orthogonal complement. This spacelike energy is related with the energy of the vector field as a map on the tangent bundle endowed with the Kaluza–Klein metric, but it is more adapted to the situation. We compute the first and second variation of the functional and we exhibit several examples of critical points on cosmological models as generalized Robertson–Walker spaces and Gödel universe, on Einstein and contact manifolds and on Lorentzian Berger’s spheres. For these critical points we have also studied to what extent they are stable or even absolute minimizers.

Trans-Planckian effects in inflationary cosmology and the modified uncertainty principle

There are good indications that fundamental physics gives rise to a modified space–momentum uncertainty relation that implies the existence of a minimum length scale. We implement this idea in the scalar field theory that describes density perturbations in flat Robertson–Walker space–time. This leads to a non-linear time-dependent dispersion relation that encodes the effects of Planck scale physics in the inflationary epoch. Unruh type dispersion relations naturally emerge in this approach, while unbounded ones are excluded by the minimum length principle. We also find red-shift induced modifications of the field theory, due to the reduction of degrees of freedom at high energies, that tend to dampen the fluctuations at trans-Planckian momenta. In the specific example considered, this feature helps determine the initial state of the fluctuations, leading to a flat power spectrum.

A field theory approach to cosmological density perturbations

Adiabatic perturbations propagate in the expanding universe like scalar massless fields in some effective Robertson–Walker space–time.

The radiation spectrum of a classical electron moving in a radiation-dominated universe

The motion and radiation of a classical electron in a conformally flat Robertson–Walker space, which corresponds to the quasi-Euclidean model of a radiation-dominated universe, is considered. The spectral-angular distribution of the energy radiated by the electron is found; the dependence of the spectrum on the momentum of the electron is investigated; estimates for the coherence interval are given. The results obtained show that the radiation spectrum of a classical electron coincides with the quasiclassical limit of the spectral distribution of the photons radiated by a Dirac electron moving in a radiation-dominated universe.

Covariant and quasi-covariant quantum dynamics in Robertson–Walker spacetimes

We propose a canonical description of the dynamics of quantum systems on a class of Robertson–Walker spacetimes. We show that the worldline of an observer in such spacetimes determines a unique orbit in the identity component SO0(4, 1) of the local conformal group of the spacetime and that this orbit determines a unique transport on the spacetime. For a quantum system on the spacetime modelled by a net of local algebras, the associated dynamics is expressed via a suitable family of ‘propagators’. In the best of situations, this dynamics is covariant, but more typically the dynamics will be ‘quasi-covariant’ in a sense we make precise. We then show, by using our technique of ‘transplanting’ states and nets of local algebras from de Sitter space to Robertson–Walker space, that there exist quantum systems on Robertson–Walker spaces with quasi-covariant dynamics. The transplanted state is locally passive, in an appropriate sense, with respect to this dynamics.

SOLUTION OF MASSLESS SPIN ONE WAVE EQUATION IN ROBERTSON–WALKER SPACE–TIME

We generalize the quantum spinor wave equation for photon into the curved space–time and discuss the solutions of this equation in Robertson–Walker space–time and compare them with the solution of the Maxwell equations in the same space–time.

Effect of Torsion in Dirac Equation for Coulomb Potential in Robertson–Walker Space–Time

The Dirac equation with Coulomb-like potential and self-interaction term, that originates from torsion, is studied in the Robertson–Walker space–time. It is shown that the angular dependence of the equation can be separated also in presence of nonlinear terms. Under reasonable physical assumptions, the time dependence is also separated. An extended perturbative calculation can then be applied qualitatively. The conclusion is that the perturbation of the energy levels of the system, as consequence of the self-interacting term, is not relevant on physical grounds.

Large extra dimensions and cosmological problems

We consider a variant of the brane-world model in which the universe is the direct product of a Friedmann, Robertson-Walker (FRW) space and a compact hyperbolic manifold of dimension $d\geq2$. Cosmology in this space is particularly interesting. The dynamical evolution of the space-time leads to the injection of a large entropy into the observable (FRW) universe. The exponential dependence of surface area on distance in hyperbolic geometry makes this initial entropy very large, even if the CHM has relatively small diameter (in fundamental units). This provides an attractive reformulation of the cosmological entropy problem, in which the large entropy is a consequence of the topology, though we would argue that a final solution of the entropy problem requires a dynamical explanation of the topology of spacetime. Nevertheless, it is reassuring that this entropy can be achieved within the holographic limit if the ordinary FRW space is also a compact hyperbolic manifold. In addition, the very large statistical averaging inherent in the collapse of the initial entropy onto the brane acts to smooth out initial inhomogeneities. This smoothing is then sufficient to account for the current homogeneity of the universe. With only mild fine-tuning, the current flatness of the universe can also then be understood. Finally, recent brane-world approaches to the hierarchy problem can be readily realized within this framework.

Geodesic connectedness and conjugate points in GRW space–times

Given two points of a generalized Robertson–Walker space–time, the existence, multiplicity and causal character of geodesics connecting them is characterized. Conjugate points of such geodesics are related to conjugate points of geodesics on the fiber, and Morse-type relations are obtained. Applications to bidimensional space–times and to GRW space–times satisfying the timelike convergence condition are also found.

A kinematic method to obtain conformal factors

Radial conformal motions are considered in conformally flat space–times and their properties are used to obtain conformal factors. The geodesic case leads directly to the conformal factor of Robertson–Walker universes. General cases admitting homogeneous expansion or orthogonal hypersurfaces of constant curvature are analyzed separately. When the two conditions above are considered together a subfamily of the Stephani perfect fluid solutions, with acceleration Fermi–Walker propagated along the flow of the fluid, follows. The corresponding conformal factors are calculated and contrasted with those associated with Robertson–Walker space–times.

PARTICLE PRODUCTION IN DE SITTER SPACE–TIME

The definition of vacuum in curved space–time is a delicate object. With Minkowski-like vacuum definition, the current has a characteristic behavior in Robertson–Walker space–time. In this work we study the behavior of Dirac current in a de Sitter space–time. The rapid oscillations of current is observed with respect to time and indicate vigorous instability in initial vacuum and is interpreted as vigorous particle production.

What can we learn from nonminimally coupled scalar field cosmology

A novel exploration of nonminimally coupled scalar field cosmology is proposedin the framework of spatially flat Friedmann—Robertson—Walker spaces forarbitrary scalar field potentials V(ψ) and values of the nonminimal couplingconstant ξ. This approach is self-consistent in the sense that the equation of stateof the scalar field is not prescribed a priori, but is rather deduced together withthe solution of the field equations. The role of nonminimal coupling appears tobe essential. A dimensional reduction of the system of differential equations leadsto the result that chaos is absent in the dynamics of a spatially flat FRW universewith a single scalar field. The topology of the phase space is studied and revealsan unexpected involved structure: according to the form of the potential V(ψ)and the value of the nonminimal coupling constant ξ, dynamically forbiddenregions may exist. Their boundaries play an important role in the topologicalorganization of the phase space of the dynamical system. New exact solutionssharing a universal character are presented; one of them describes a nonsingularuniverse that exhibits a graceful exit from, and entry into, inflation. This behaviordoes not require the presence of the cosmological constant. The relevance of thissolution and of the topological structure of the phase space with respect to anemergence of the universe from a primordial Minkowski vacuum, in an extendedsemiclassical context, is shown.