Encoding Possible Final States of the Universe with Conformal Structures

Abstract

The concept of an Isotropic (Past) Singularity (IPS) was defined by Goode and Wainwright in 1985 as a mathematical formalisation of quiescent cosmology and the Weyl Curvature Hypothesis (WCH) for the isotropic initial state of the universe.

¶ In this thesis it is argued that the framework of an IPS is not sufficient to guarantee a future behaviour which is compatible with the future anisotropy implied by quiescent cosmology and the WCH. Therefore it is necessary to complete and combine the framework of an IPS with new definitions, in order to assure an appropriate past and future behaviour of a cosmology satisfying the respective combination of definitions. Since it is not yet clear whether our universe will expand indefinitely or recontract, it is reasonable to provide a new definition for the scenario of an ever expanding cosmos and one for a recollapsing universe.

¶ Specific example space-times are explored for their conformal structure, future evolution and compatibility with the WCH as guidance in the quest for the new definitions. Motivated by these particular models, we present for the first time the definitions for the conformal structure of an Anisotropic Future Endless Universe (AFEU) and an Anisotropic Future Singularity (AFS). For the purpose of completeness and comparison, we furthermore define the physically less realistic Isotropic Future Singularity (IFS) and the Future Isotropic Universe (FIU).

¶ A number of essential technical implications of the new de nitions are derived. It is explicitly shown that a conformal structure, whose conformal factor is a function of cosmic time, necessarily leads to an asymptotically Ricci dominated Weyl curvature and asymptotically expansion dominated kinematics, if the conformal metric remains regular. This condition is satis ed by the IFS and FIU. Based on this, it is argued that a conformal structure for an anisotropic nal state of the universe requires a degenerate conformal metric, as is the case for the AFS and AFEU. ¶ This degeneracy complicates the derivation of physical attributes of the concepts of an AFEU and an AFS and, consequently, new approaches are unavoidable. Some physical properties are examined, such as the behaviour of the expansion scalar and the curvature. It is proven that the conformal space-times always possess a future singularity, which under reasonable assumptions corresponds to a strong curvature singularity. Finally, we reveal sufficient conditions for the AFS, as well as the IPS, to be a strong curvature singularity. ¶ The combination of the IPS with the AFEU and the AFS could provide a possible first version of a complete mathematical formalisation of quiescent cosmology. Show more