103. Zdzislaw Pogoda, Leszek M. Sokołowski
Does Mathematics Distinguish Certain Dimensions of Spaces? Part I.
The American Mathematical Monthly, vol. 104, pp. 860-869 (1997).
[journal] | Abstract:
Abstract | 104. Marco Litterio, Leszek M. Sokołowski, Zdzisław A. Golda, Luca Amendola, Andrzej Dyrek
Anisotropic inflation from extra dimensions
Phys. Lett., vol. B382, pp. 45-52 (1996).
[abstract] [preprint] [journal] | Abstract:
Vacuum multidimensional cosmological models with internal spaces being compact $n$-dimensional Lie group manifolds are considered. Products of 3-spheres and $SU(3)$ manifold (a novelty in cosmology) are studied. It turns out that the dynamical evolution of the internal space drives (power-law) inflation in the external world. This inflationary solution brings two extra bonuses: 1) it is an attractor in phase space (no fine-tuning required); 2) it is determined by the Lie group space solely and not by any arbitrary inflaton potentials. Only scalar fields representing the anisotropic scale factors of the internal space appears in the four dimensions. The size of the volume of the internal space at the end of inflation is compatible with observational constraints. This simple and natural model can be completed by some extended-inflation-like mechanism that ends the inflationary evolution. | 105. Marek Szydłowski, Michael Heller, and Wiesław Sasin
Geometry of spaces with the Jacobi metric
J. Math. Phys., vol. 37, p. 346 (1996).
[abstract] [journal] | Abstract:
The generalized Maupertuis principle is formulated for systems with the natural Lagrangian and an indefinite form of the kinetic energy. The generalization is applied to the theory of gravity and cosmology. For such systems, the metric determined by the kinetic energy form has a Lorentz signature. The theorem is proved concerning the behavior of trajectories in a neighborhood of the boundary of the region admissible for motion. This region is not a smooth manifold but turns out to be a differential space of constant differential dimension. This fact allows us to use geometric methods analogous to those elaborated for smooth manifolds. It is shown that singularities of the Jacobi metric are not dangerous for the motion; its trajectories are smooth in the sense of the theory of differential spaces. | 106. Michael Heller and Wiesław Sasin
Noncommutative structure of singularities in general relativity
J. Math. Phys., vol. 37, p. 5665 (1996).
[abstract] [journal] | Abstract:
Initial and final singularities in the closed Friedman world model are typical examples of malicious singularities. They form the single point of Schmidt’s b‐boundary of this model and are not Hausdorff separated from the rest of space–time. The method of noncommutative geometry, developed by A. Connes and his co‐workers, is applied to this case. We rephrase Schmidt’s construction in terms of the groupoid math of orthonormal frames over space–time and carry out the ‘‘desingularization’’ process. We define the line bundle τ:Ω1/2→math over math and change the space of its cross sections into an involutive algebra. This algebra is represented in the space of operators on a Hilbert space and, with the norm inherited from these operators, it becomes a C∗‐algebra. The initial and final singularities of the closed Friedman model are given by two distinct representations of this C∗‐algebra in the space of operators acting on the Hilbert space L2(O(3,1)). | 107. Zdzisław A. Golda, Marco Litterio, Leszek M. Sokołowski, Luca Amendola, Andrzej Dyrek
Pure Geometrical Evolution of the Multidimensional Universe
Ann. Phys. , vol. 248, pp. 246-285 (1996).
[abstract] [journal] | Abstract:
An exhaustive qualitative analysis of cosmological evolution for some multidimensional universes is given. The internal space is taken to be a compact Lie group Riemannian manifold. The space is generically anisotropic; i.e., its cosmological evolution is described by its (time-dependent) volume, the dilaton, and by relative anisotropic deformation factors representing the shear of the internal dimensions during the evolution. Neither the internal space nor its subspaces need to be Einstein spaces. The total spacetime is empty, and the cosmic evolution of the external, four-dimensional world is driven by the geometric ``matter'' consisting of the dilaton and of the deformation factors. Since little is known about any form of matter in the extra dimensions, we do not introduce any {\it ad hoc\/} matter content of the Universe. We derive the four-dimensional Einstein field equations (with a cosmological term) for these geometric sources in full generality, i.e., for any compact Lie group. A detailed analysis is done for some specific internal geometries: products of 3-spheres, and $SU$(3) space. Asymptotic solutions exhibit power law inflation along with a process of full or partial isotropization. For the $SU$(3) space all the deformation factors tend to a common value, whereas in the case of $S^3$'s each sphere isotropizes separately. | 108. David H Lyth, Andrzej Woszczyna
Large scale perturbations in the open universe
Phys. Rev. D, vol. 52, pp. 3338-3357 (1995).
[abstract] [preprint] [journal] | Abstract:
When considering perturbations in an open (Omega<1) universe, cosmologists retain only sub-curvature modes (defined as eigenfunctions of the Laplacian whose eigenvalue is less than -1 in units of the curvature scale, in contrast with the super-curvature modes whose eigenvalue is between -1 and 0). Mathematicians have known for almost half a century that all modes must be included to generate the most general HOMOGENEOUS GAUSSIAN RANDOM FIELD, despite the fact that any square integrable FUNCTION can be generated using only the sub-curvature modes. The former mathematical object, not the latter, is the relevant one for physical applications. The mathematics is here explained in a language accessible to physicists. Then it is pointed out that if the perturbations originate as a vacuum fluctuation of a scalar field there will be no super-curvature modes in nature. Finally the effect on the cmb of any super-curvature contribution is considered, which generalizes to Omega<1 the analysis given by Grishchuk and Zeldovich in 1978. A formula is given, which is used to estimate the effect. In contrast with the case Omega=1, the effect contributes to all multipoles, not just to the quadrupole. It is important to find out whether it has the same l dependence as the data, by evaluating the formula numerically.
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