The supersymmetric Casimir effect and quantum creation of the universe with nontrivial topology

We estimate the probability of quantum creation of the universe, having the spatial topology (S¹)³, and filled with the fields of minimal N = 1 supergravity, in the semiclassical approximation. After creation, inflation of the universe occurs due to the topological Casimir effect. Creation of the universe with an isotropic topology is found to be the most preferable.     

On the Birth of a Closed Hyperbolic Universe

We clarify and develop the results of a previous paper on the birth of a closed universe of negative spatial curvature and multiply connected topology. In particular we discuss the initial instanton and the second topology change in more detail. This is followed by a short discussion of the results.     

What do we know and what can we learn about the topology of the universe?

We review the main mathematical concepts of cosmic topology and the main observational methods to study the topology of the universe. We then show how topology can play a crucial role in the early universe.     

A cyclic universe with colour fields

The topology of the universe is discussed in relation to the singularity problem. We explore the possibility that the initial state of the universe might have had a structure with 3-Klein bottle topology, which would lead to a model of a nonsingular oscillating (cyclic) universe with a well-defined boundary condition. The same topology is assumed to be intrinsic to the nature of the hypothetical primitive constituents of matter (usually called preons) giving rise to the observed variety of elementary particles. Some phenomenological implications of this approach are also discussed.      

The topology of superclusters: A window on the early universe

Analysis of the topology of the distribution of matter in the universe provides information not found in correlation functions. Quantitative measures of the topology can be used to test the hypothesis that structure in our universe grew by gravitational amplification of Gaussian fluctuations.     

Topology of the Universe: Background and recent observational approaches

Is the Universe (a spatial section thereof) finite or infinite? Knowing the global geometry of a Friedmann-Lemaître (FL) universe requires knowing both its curvature and its topology. A flat or hyperbolic (‘open’) FL universe is not necessarily infinite in volume. Multiply connected flat and hyperbolic models are, in general, as consistent with present observations on scales of 1–20 h^?1 Gpc as are the corresponding simply connected flat and hyperbolic models. The methods of detecting multiply connected models (MCM’s) are presently in their pioneering phase of development and the optimal observationally realistic strategy is probably yet to be calculated. Constraints against MCM’s on ～1–4 h^?1 Gpc scales have been claimed, but relate more to inconsistent assumptions on perturbation statistics rather than just to topology. Candidate 3-manifolds based on hypothesised multiply imaged objects are being offered for observational refutation. The theoretical and observational sides of this rapidly developing subject have yet to make any serious contact, but the prospects of a significant detection in the coming decade may well propel the two together.     

Analysis of the Topology of the Kerr Metric

The equations of motion of a test particle are integrated for the field of a rotating Kerr black hole (BH) (in accordance with [1]). Due to the lack of analytical transformations for the Carter–Penrose diagrams (CPDs) for the Kerr metric, the topology of the Kerr BH is studied by analytical investigation of the equations of motion. Transformations for the CPDs for the Reisner–Nordström metric are analyzed. The problem of boundary conditions for the Reisner–Nordström topology is analyzed. A solution to this problem of boundary conditions is proposed. It is proved that, in the Reisner–Nordström topology, only one way to go to another universe is possible. For the Kerr topology, the possibility of the existence of an alternative transition to another universe that does not coincide with the universe for the ordinary transition is found. This alternative transition is performed through a surface with a zero radial coordinate (zero radius). Initial conditions for the falling particle are found that correspond to an alternative transition to another universe. The tidal forces acting on a falling body in the Kerr metric are estimated, and the possibility of the transition of the body to other universes without being destroyed by tidal forces is proved.     

The premature recollapse problem in closed inflationary universes

We discuss whether closed universe can avoid recollapsing before inflation ensues. We show that in general closed universe are not equivalent to recollapsing universes or positive curvature universes. Closed universes will not in general recollapse if the matter content violates the strong energy condition. This violation is also a necessary condition for inflation to occur. When the strong energy condition holds closed universes can only recollapse if they possess S³ or S²×S¹ spatial topology. Even when the topology is S³ and the strong energy condition holds it is not known whether anisotropic closed universes do all recollapse. We give examples to show that closed universes which begin in an extremely anisotropic state cannot recollapse until they are close to isotropy. This suggests that if the initial conditions prior to inflation are sufficiently anisotropic then the universe cannot recollapse until it has been isotropized by inflation. We also discuss the existence of inflation in isotropic cosmological models in R+R² lagrangian theories of gravity and extend a result of Whitt to show that such theories are conformally equivalent to general relativity plus a scalar field with an asymmetric potential.     

Dimensional reduction in an oscillatory Kaluza-Klein cosmology

We study the dynamical evolution of a seven-dimensional, homogeneous, cosmological model. Anisotropy is prevailed all over the spatial dimensions with the topology S³ × S³, and leads to repetitious (nonchaotic) oscillations of the scale factors. The universe evolves from an initial Kasner phase to a final one, from which the dimensional reduction results. We pay attention to the effect of radiation-energy density and discuss the isotropization of ordinary dimensions through the stabilization of extra dimensions at the Kaluza-Klein radius.     

Constraining the topology of the universe

The first year data from the Wilkinson Microwave Anisotropy Probe are used to place stringent constraints on the topology of the Universe. We search for pairs of circles on the sky with similar temperature patterns along each circle. We restrict the search to back-to-back circle pairs, and to nearly back-to-back circle pairs, as this covers the majority of the topologies that one might hope to detect in a nearly flat universe. We do not find any matched circles with radius greater than 25 degrees. For a wide class of models, the nondetection rules out the possibility that we live in a universe with topology scale smaller than 24 Gpc.     

A Closed Contour of Integration in Regge Calculus

The analytic structure of the Regge action on a cone in d dimensions over a boundary of arbitrary topology is determined in simplicial minisuperspace. The minisuperspace is defined by the assignment of a single internal edge length to all 1-simplices emanating from the cone vertex, and a single boundary edge length to all 1-simplices lying on the boundary. The Regge action is analyzed in the space of complex edge lengths, and it is shown that there are three finite branch points in this complex plane. A closed contour of integration encircling the branch points is shown to yield a convergent real wave function. This closed contour can be deformed to a steepest descent contour for all sizes of the bounding universe. In general, the contour yields an oscillating wave function for universes of size greater than a critical value which depends on the topology of the bounding universe. For values less than the critical value the wave function exhibits exponential behaviour. It is shown that the critical value is positive for spherical topology in arbitrary dimensions. In three dimensions we compute the critical value for a boundary universe of arbitrary genus, while in four and five dimensions we study examples of product manifolds and connected sums.     

Can the CMB reveal the topology of the universe?

This article summarizes recent progress in the development of tools to study the topology of the universe with the cosmic microwave background. The different signatures of the topology and observational constraints are described. The ability of future experiments to reveal the topological structure of our universe is then discussed. To cite this article: J.P. Uzan, A. Riazuelo, C. R. Physique 4 (2003).     

Dark energy,curvature, and cosmic coincidence

The fact that the energy densities of dark energy and matter are similar currently, known as the coincidence problem, is one of the main unsolved problems of cosmology. We present here a model in which a spatial curvature of the universe can lead to a transition in the present epoch from a matter dominated universe to a scaling dark energy dominance in a very natural way. In particular, we show that if the exponential potential of the dark energy field depends linearly on the spatial curvature density of a closed universe, the observed values of some cosmological parameters can be obtained assuming acceptable values for the present spatial curvature of the universe, and without fine tuning in the only parameter of the model. We also comment on possible variations of this model, and realistic scenarios in which it could arise.     

Living with phantoms fields in a sheet spacetime

We consider the flat anisotropic Bianchi I braneworld model of the universe within the framework of low energy effective string action in four-dimensions including the leading order α′ terms, two-scalar fields, their interaction, non-minimal coupling of the dark-energy scalar field to the scalar curvature and effective cosmological constant. Backward (high energy limit) and forward (low energy limit) in time analytic solutions are derived and late-time accelerated expansion was found. It is shown that during the transition from high energy limit to the low energy limit, the topology of the universe is changing in time: we have a transition from a (1 + 3) FRW homogenous and isotropic spacetime dominated by radiation to a (1 + 2) spacetime sheet dominated by phantom energy while the third spatial dimension is contracted in time. We have also found that dark matter and dark energy may be unified at early epoch in the form of radiation fluids while the late-time dynamics is governed by phantom energy and dark energy. Many interesting features are revealed.     

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.     

The energy of the Universe

References to energy of the universe have focussed upon the matter contribution, whereas the conservation laws must include a gravitational contribution as well. The conservation laws as applied to FRW cosmologies suggest a zero total energy irrespective of the spatial curvature when the value of the cosmological constant is taken to be zero. This result provides a useful constraint on models of the early universe and lends support to currently studied theories of the universe arising as a quantum fluctuation of the vacuum.     

A two-mass expanding exact space-time solution

In order to understand how locally static configurations around gravitationally bound bodies can be embedded in an expanding universe, we investigate the solutions of general relativity describing a space–time whose spatial sections have the topology of a 3-sphere with two identical masses at the poles. We show that Israel junction conditions imply that two spherically symmetric static regions around the masses cannot be glued together. If one is interested in an exterior solution, this prevents the geometry around the masses to be of the Schwarzschild type and leads to the introduction of a cosmological constant. The study of the extension of the Kottler space–time shows that there exists a non-static solution consisting of two static regions surrounding the masses that match a Kantowski–Sachs expanding region on the cosmological horizon. The comparison with a Swiss-Cheese construction is also discussed.     

The energy localization problem and the renormalized vacuum energy in static Robertson-Walker universes

We calculate the renormalized quantum vacuum energy inside a spherical boundary for the massless conformal scalar field in curved background Robertson-Walker geometry. We use the mode sum method with an exponential cutoff. In our calculations we do not make assumptions about the exterior geometry or the global topology of the universe.     

Cosmological textures

The notions of phase transitions and causality, combined with the standard cosmological model, lead to the appearance of topological defects in the early universe. The most familiar types of defects are solitons, strings, and domain walls. Another type can exist when the spatial universe is compact. When these appear the whole universe takes on a winding number, and the consequences are quite amusing; for example, it is possible that a closed universe can mimic open or flat universes. Another possibility is that the vacuum has a nonabelian magnetic field strength at all points in the universe.This essay received the fifth award from the Gravity Research Foundation for the year 1986-Ed.Work supported by the Department of Energy, contract DE-AC03-76SF00515.