Integrable Multicomponent Perfect Fluid Multidimensional Cosmology II: Scalar Fields

We consider anisotropic cosmological models with a universe of dimension 4 or higher, factorized into n 2 Ricci-flat spaces, containing an m-component perfect fluid of m non-interacting homogeneous minimally coupled scalar fields under special conditions. We describe the dynamics of the universe: It has a Kasner-like behaviour near the singularity and isotropizes during the expansion to infinity. Some of the models considered are integrable, and classical as well as quantum solutions are found. Some solutions produce inflation from nothing. There exist classical asymptotically anti-de Sitter wormholes, and quantum wormholes with discrete spectrum.     

A Quantum Cosmological Model with Static and Dynamic Wormholes

Quantization is performed for a Friedmann-Robertson-Walker universe filled with a conformally invariant scalar field and a perfect fluid with equation of state p = . A well-known discrete set of static quantum wormholes is shown to exist for radiation ( = 1/3), and a novel continuous set is found for cosmic strings ( = –1/3), the latter states having throat radii of any size. In both cases wave-packet solutions to the Wheeler-DeWitt equation are obtained with all the properties of evolving quantum wormholes. In the case of a radiation fluid, a detailed analysis of the quantum dynamics is made in the context of the Bohm-de Broglie interpretation. It is shown that a repulsive quantum force inversely proportional to the cube of the scale factor prevents singularities in the quantum domain. For the states considered, there are no particle horizons either.     

Can semi-classical wormholes solve the cosmological horizon problem?

Measurements of the cosmic microwave background (cmb) radiation provide the strongest evidence for the isotropy of the observable universe on the largest scales. However, thecmb is received from regions which were not in causal contact at the time of last scattering. This is the horizon problem and the generally accepted solution is to invoke an inflationary period in the early universe. We consider the possibility that the universe did not necessarily inflate, but was filled with a network of evolving wormholes connecting otherwise causally disjoint regions. These wormholes emerged naturally from the Planck epoch and need only have stayed open for a very brief time (t<10^–34 sec) in order to have thermalized the early universe.     

Bimetric general relativity and cosmology

A modification of the general relativity theory is proposed (bimetric general relativity) in which, in addition to the usual metric tensorg _v describing the space-time geometry and gravitation, there exists also a background metric tensor _v The latter describes the space-time of the universe if no matter were present and is taken to correspond to a space-time of constant curvature with positive spatial curvature (k=1). Field equations are obtained, and these agree with the Einstein equations for systems that are small compared to the size of the universe, such as the solar system. Energy considerations lead to a generalized form of the De Donder condition. One can set up simple isotropic closed models of the universe which first contract and then expand without going through a singular state. It is suggested that the maximum density of the universe was of the order ofc ^{5 –1} G ^–210⁹³ g/cm³. The expansion from such a high-density state is similar to that from the singular state (big bang) of the general relativity models. In the case of the dust-filled model one can fit the parameters to present cosmological data. Using the radiation-filled model to describe the early history of the universe, one can account for the cosmic abundance of helium and other light elements in the same way as in ordinary general relativity.     

Homogeneous Kähler manifolds andT-algebras inN=2 supergravity and superstrings

Motivated by the problem of the moduli space of superconformal theories, we classify all the (normal) homogeneous Kähler spaces which are allowed in the coupling of vector multiplets toN=2 SUGRA. Such homogeneous spaces are in one-to-one correspondence with the homogeneous quaternionic spaces (H ⁿ ) found by Alekseevskii. There are two infinite families of homogeneous non-symmetric spaces, each labelled by two integers. We construct explicitly the corresponding supergravity models. They are described by acubic functionF, as in flat-potential models. They are Kähler-Einstein if and only if they are symmetric. We describe in detail the geometry of the relevant manifolds. They are Siegel (bounded) domains of the first type. We discuss the physical relevance of this class of bounded domains for string theory and the moduli geometry. Finally, we introduce theT-algebraic formalism of Vinberg to describe in an efficient way the geometry of these manifolds. The homogeneous spaces allowed inN=2 SUGRA are associated to rank 3T-algebras in exactly the same way as the symmetric spaces are related to Jordan algebras. We characterize theT-algebras allowed inN=2 supergravity. They are those for which theungraded determinant is a polynomial in the matrix entries. The Kähler potential is simply minus the logarithm of this naive determinant.     

Asymptotic Self-similarity Breaking in Cosmology

We discuss the role of self-similarity in the evolution of cosmological models. The simplest model, the flat Friedmann–Lematre universe is exactly self-similar. On the other hand, the open Friedmann–Lematre universe and the anisotropic Bianchi I universes, are not exactly self-similar, but are asymptotically self-similar, both near the initial singularity and at late times. In general, however, cosmological models are not asymptotically self-similar, and our goal is to describe the different mechanisms that lead to asymptotic self-similarity breaking in Bianchi universes. The discussion will also serve to give an overview of our current understanding of the dynamics of Bianchi universes.     

Nonvacuum taub-type cosmological model

The Einstein universe is a simple model describing a static cosmological spacetime, having a constant radius and a constant curvature, and, as is well known, it does not describe our universe. We propose a model which is an extension of Einstein's. Our metric, havingR × S ³ topology, describes a nonisotropic homogeneous closed (finite) universe of Bianchi type IX. This metric is similar to that of Taub, but is simpler. Unlike the Taub solution (which is a cosmological extension of the NUT solution), however, the universe described by our metric contains matter. Like the Taub metric, our metric has two positive constants (, T). The gravitational red shift calculated from our metric is given. Similarly to the Schwarzschild metric, which has a singularity atr = 2m, this metric has the same kind of singularity att = 2. The maximal extension of the coordinates in our metric is fairly analogous to that of the Schwarzschild metric.     

The null energy condition in wormholes with cosmological constant

Dynamic and static wormhole solutions of Einsteins equations with the cosmological constant are presented. The dynamic solutions can be interpreted as Friedmann–Robertson–Walker models with traversable wormholes. The null energy condition is checked for both dynamic and static wormholes and it is shown explicitly that the cosmological constant modifies the violation of this condition.     

Wormholes, Classical Limit and Dynamical Vacuum in Quantum Cosmology

First a Friedmann-Robertson-Walker (FRW)universe filled with dust and a conformally invariantscalar field is quantized. For the closed model we finda discrete set of wormhole quantum states. In the case of flat spacelike sections we find states withclassical behaviour at small values of the scale factorand quantum behaviour for large values of the scalefactor. Next we study a FRW model with a conformally invariant scalar field and a nonvanishingcosmological constant dynamically introduced byregarding the vacuum as a perfect fluid with equation ofstate p = –. The ensuing Wheeler-DeWittequation turns out to be a bona fide Schrodinger equation, andwe find that there are realizable states with a definitevalue of the cosmological constant. Once again we findfinite-norm solutions to the Wheeler-DeWitt equation with definite values of thecosmological constant that represent wormholes,suggesting that in quantum cosmological models with asimple matter content wormhole states are a commonoccurrence.     

Kinematical properties of topologically nontrivial models of space-time

Using simple geometric models of wormholes as examples, we analyze the influence of the boundary conditions arising as a result of sewing the inner and outer spaces, on the casual structure of space-time and clock synchronization. It is shown that the relativity principle cannot be applied to the motion of the wormhole in the outer space. We demonstrate that it is impossible to dynamically transform a wormhole into a time machine. It is noted that the considered models are counterexamples to a number of statements concerning causality violation.VMTs Integral. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 83–88, December, 1992.     

Accelerated expansion from braneworld models with variable vacuum energy

We study the acceleration of the universe as a consequence of the time evolution of the vacuum energy in cosmological models based in braneworld theories in 5D. A variable vacuum energy may appear if the size of the extra dimension changes during the evolution of the universe. In this scenario the acceleration of the universe is related not only to the variation of the cosmological term, but also to the time evolution of G and, possibly, to the variation of other fundamental constants as well. This is because the expansion rate of the extra dimensionappears in different contexts, notably in expressions concerning the variation of rest mass and electric charge. We concentrate our attention on spatially-flat, homogeneous and isotropic, brane-universes where the matter density decreases as an inverse power of the scale factor, similar (but at different rate) to the power law in FRW-universes of general relativity. We show that these braneworld cosmologies are consistent with the observed accelerating universe and other observational requirements. In particular, G becomes constant and asymptotically in time. Another important feature is that the models contain no adjustable parameters. All the quantities, even the five-dimensional ones, can be evaluated by means of measurements in 4D. We provide precise constrains on the cosmological parameters and demonstrate that the effective equation of state of the universe can, in principle, be determined by measurements of the deceleration parameter alone. We give an explicit expression relating the density parameters , and the deceleration parameter q. These results constitute concrete predictions that may help in observations for an experimental/observational test of the model.     

Observational effects in black and white holes (dynamical wormholes)

The models of wormholes with a topology based on a Reissner-Nordström black and white hole are considered. In these models, there are one entrance in one universe (a black hole) and one exit into another universe (a white hole) corresponding to this entrance. The passage of matter through the wormhole in these models is possible only in one direction (from past to future). All models are considered under the assumption of spherical symmetry. It is shown that all models without a throat do not violate the null energy condition. The model of a Reissner-Nordström black hole containing no singularities inside the horizon has been constructed. The trajectories of particles and light rays passing from one universe into another have been constructed for the simplest Reissner-Nordström black and white hole. Distinctive features have been found for the images of objects from another universe observed through such objects. The characteristics of these images are compared with those for ordinary wormholes.     

Comment on a paper by G. H. Weiss,S. Havlin,and A. Bunde

A simple proof is pointed out for the asymptotic exponential decay of then-step survival probability of a random walk on a finite lattice with traps in the limit asn . Some bounds are mentioned, which are valid for finiten and for symmetric random walks.     

Scalar Fields in Multidimensional Gravity. No-Hair and Other No-Go Theorems

Global properties of static, spherically symmetric configurations with scalar fields of sigma-model type with arbitrary potentials are studied in D dimensions, including models where the space-time contains multiple internal factor spaces. The latter are assumed to be Einstein spaces, not necessarily Ricci-flat, and the potential V includes a contribution from their curvatures. The following results generalize those known in four dimensions: (A) a no-hair theorem on the nonexistence, in case V 0, of asymptotically flat black holes with varying scalar fields or moduli fields outside the event horizon; (B) nonexistence of particlelike solutions in field models with V 0; (C) nonexistence of wormhole solutions under very general conditions; (D) a restriction on possible global causal structures (represented by Carter-Penrose diagrams). The list of structures in all models under consideration is the same as is known for vacuum with a cosmological constant in general relativity: Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild – de Sitter, and horizons which bound a static region are always simple. The results are applicable to various Kaluza-Klein, supergravity and stringy models with multiple dilaton and moduli fields.     

Selection rules for topology change

It is shown that there are restrictions on the possible changes of topology of space sections of the universe if this topology change takes place in a compact region which has a Lorentzian metric and spinor structure. In particular, it is impossible to create a single wormhole or attach a single handle to a spacetime but it is kinematically possible to create such wormholes in pairs. Another way of saying this is that there is a ₂ invariant for a closed oriented 3-manifold which determines whether can be the spacelike boundary of a compact manifoldM which admits a Lorentzian metric and a spinor structure. We evaluate this invariant in terms of the homology groups of and find that it is the mod2 Kervaire semi-characteristic.     

Maximal variety as a new fundamental principle of dynamics

It is suggested, following a proposal made recently by Smolin, that the most fundamental law of the universe takes this form: Among the set of all possible universes compatible with an irreducibly minimal set of structural constraints, the actually realized universe is the one which maximizes a mathematically well-defined number (the variety) that measures the structural variety of the universe (in the totality of its history). This gives expression to Leibniz's idea that the actual universe gives the greatest variety possible, but with the greatest possible order. Two models are proposed in which the idea can be realized and its consequences tested; both are discrete in nature and satisfy highly nonlocal laws. In such a scheme a unique (finite) universe is called into being by the fundamental requirement of maximal variety (for given definition of the variety), which it is conjectured could have such a powerful ordering effect that space, time, the currently known laws of physics, and the observed structure of the universe could all appear as emergent consequences of the single underlying law.Invited contribution honoring Professor Peter Mittelstaedt on the occasion of his 60th birthday.     

First-Order Intertwining Operators with Position Dependent Mass and <Emphasis Type="Italic">η</Emphasis>-Weak-Pseudo-Hermiticity Generators

A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of η-weak-pseudo-Hermitian position-dependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target η-weak-pseudo-Hermitian PDM—Hamiltonians’ map is suggested. Some η-weak-pseudo-Hermitian -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-levels crossing and flown-away states phenomena are reported for the resulting Scarf II spectrum. Some of the corresponding η-weak-pseudo-Hermitian Scarf II- and periodic-type-isospectral models ( -symmetric and non- -symmetric) are given as products of the reference-target map.     

Three-pion resonances

We investigate the continuum three-pion problem within a relativistic three-body model that takes into account the S andP waves. The dynamical input of the two-body subsystem is given by separable potentials, which yield a good fit to the scattering data and resonance parameters up to a two-body invariant mass of 900MeV. We introduce a parameter expressing the ambiguity in the reduction of a fully relativistic theory to a three-dimensional one. The masses and widths of the ,a ₁(1260), and (1300) mesons, which decay predominantly into three pions, are reasonably well described by our model. Theh ₁(1170) meson, however, which also decays into three pions, cannot be explained as a three-pion resonance. Some Argand diagrams are shown in those channels where resonances exist.     

Cosmological models in a Kaluza-Klein theory with variable rest mass

A general class of solutions is obtained for a homogeneous, spatially isotropic five-dimensional (5D) Kaluza-Klein theory with variable rest mass. These solutions generalize in the algebraic and physical sense the previously found solutions in the literature. The 4D spacetime sections of the solutions reduce to the Minkowski metric, K=0 Robertson-Walker metric with the equation of statep=np (p=pressure,n=constant sound speed,=energy density), and to the Robertson-Walker spacetime with steady-state metric. Some of the solutions, in different limits, show compactification of the fifth dimension. Some extensions of the model are discussed.     

Warped compactification to de Sitter space

We explore in detail the prospects of obtaining a four-dimensional de Sitter universe in classical supergravity models with warped and time-independent extra dimensions, presenting explicit cosmological solutions of the (4+n)$(4+n)$-dimensional Einstein equations with and without a bulk cosmological constant term. For the first time in the literature we show that there may exist a large class of warped supergravity models with a noncompact extra dimension which lead to a finite 4D Newton constant as well as a massless 4D graviton localized on an inflating four-dimensional FLRW universe. This result helps establish that the ‘no-go’ theorem forbidding acceleration in ‘standard’ compactification of string/M-theory on physically compact spaces should not apply to a general class of warped supergravity models that allows at least one noncompact direction. We present solutions for which the size of the radial dimension takes a constant value in the large volume limit, providing an explicit example of spontaneous compactification.