Spiros Cotsakis

Inflation and the Conformal Structure of Higher Order Gravity Theories

Abstract We examine gravity theories derived from a gravitational lagrangian that is and analytic function ƒ(R) of the scalar curvature R in a space-time of arbitrary dimension D. We show that they are conformally equivalent to general relativity plus a scalar-field matter source with a particular self-interaction potential. The general form of this potential is calculated to be V(o) = 1 2[Rƒ′(R)−ƒ(R)][ƒ′(R)] D (2−D) , where exp (o) = [ƒ′ (R)] 2 (D−2) . Flat potentials arise as o→+∞ for polynomial lagrangians of leading order β whenever D = 2β. Several explicit examples are given and discussed with reference to inflation. We show how our results can lead to singularity theorems for gravity theories derived from a general lagrangian.

Future singularities of isotropic cosmologies

Abstract We show that globally and regularly hyperbolic future geodesically incomplete isotropic universes, except for the standard all-encompassing ‘big crunch’, can accommodate singularities of only one kind, namely, those having a non-integrable Hubble parameter, H. We analyze several examples from recent literature which illustrate this result and show that such behaviour may arise in a number of different ways. We also discuss the existence of new types of lapse singularities in inhomogeneous models, impossible to meet in the isotropic ones.

Global Hyperbolicity and Completeness

We prove global hyperbolicity of spacetimes under generic regularity conditions on the metric. We then show that these spacetimes are timelike and null geodesically complete if the gradient of the lapse and the extrinsic curvature K are integrable. This last condition is required only for the tracefree part of K if the universe is expanding.

The Dominant balance at cosmological singularities

We define the notion of a finite-time singularity of a vector field and then discuss a technique suitable for the asymptotic analysis of vector fields and their integral curves in the neighborhood of such a singularity. Having in mind the application of this method to cosmology, we also provide an analysis of the time singularities of an isotropic universe filled with a perfect fluid in general relativity.

Symmetry, Singularities and Integrability in Complex Dynamics I: The Reduction Problem

Abstract Quadratic systems generated using Yang-Baxter equations are integrable in a sense, but we display a deterioration in the possession of the Painlevé property as the number of equations in each ‘integrable system’ increases. Certain intermediate systems are constructed and also tested for the Painlevé property. The Lie symmetries are also computed for completeness.

Self-regenerating inflationary universe in higher-order gravity in arbitrary dimension

Abstract We present a discussion of self-regenerating, self-reproducing inflationary universe in the context of higher-order gravity theories in an arbitrary number of dimensions taking into account quantum fluctuations during the time of inflation. Using the conformal equivalence theorem, we analyse the scalar field dynamics induced by quantum fluctuations in an effective potential which is typically an exponential function of the field. We find that, as in the usual eternal inflation model based on chaotic inflation, space-time foam regions are likely to exist today for some choices of the gravitational lagrangian. These include all theories with space-time dimension greater than twice the order of the leading term in the gravitational lagrangian. However, when the number of space-time dimensions is less or equal to twice the order of the leading term in the lagrangian, eternal inflation does occur and space-time foam regions are not likely to exist today.

A general sudden cosmological singularity

We construct an asymptotic series for a general solution of the Einstein equations near a sudden singularity. The solution is quasi isotropic and contains nine independent arbitrary functions of the space coordinates as required by the structure of the initial value problem.

Symmetry, Singularities and Integrability in Complex Dynamics V: Complete Symmetry Groups of Certain Relativistic Spherically Symmetric Systems

Abstract We show that the concept of complete symmetry group introduced by Krause (J. Math. Phys. 35 (1994), 5734–5748) in the context of the Newtonian Kepler problem has wider applicability, extending to the relativistic context of the Einstein equations describing spherically symmetric bodies with certain conformal Killing symmetries. We also provide a simple demonstration of the nonuniqueness of the complete symmetry group.

Mathematical and Quantum Aspects of Relativity and Cosmology

Global Wave Maps on Curved Space Times.- Einsteins Equations and Equivalent Hyperbolic Dynamical Systems.- Generalized Bowen-York Initial Data.- The Reduced Hamiltonian of General Relativity and the ?-Constant of Conformal Geometry.- Anti-de-Sitter Spacetime and Its Uses.- Black Holes and Wormholes in 2+1 Dimensions.- Open Inflation.- Generating Cosmological Solutions from Known Solutions.- Multidimensional Cosmological and Spherically Symmetric Solutions with Intersecting p-Branes.- Open Issues.

Generalized cosmic no-hair theorems

We show that the quasi-exponential solution exp (Bt−At2) of R+αR2 gravity is an attractor for all homogeneous and isotropic solutions of higher order gravity theories derived from a lagrangian that is an arbitrary analytic function of the scalar curvature R. This indicates that a generalized form of the cosmic no-hair conjecture is valid in the framework of higher order gravity theories.