- The paper demonstrates that cyclic models using the IS mechanism for entropy dilution remain geodesically incomplete.
- The authors apply the Borde-Guth-Vilenkin theorem to show that a net positive average Hubble parameter confines past proper time, limiting cosmic extendibility.
- The analysis highlights that while the periodic Hubble parameter successfully manages entropy, it simultaneously introduces challenges for achieving a truly infinite temporal framework.
Cyclic Cosmology and Geodesic Completeness
The paper "Cyclic Cosmology and Geodesic Completeness" by William H. Kinney and Nina K. Stein investigates the theoretical underpinnings of bouncing cosmological models, specifically focusing on the geodesic completeness of such universes. Cyclic models have gained attention as an attractive alternative to cosmological inflation, proposing an oscillating universe that expands and contracts periodically, thereby eliminating the need for a primordial boundary condition like the big bang singularity.
The authors critically assess the recently proposed Ijjas-Steinhardt (IS) model, which attempts to resolve entropy issues inherent in cyclic cosmology through a scale factor that grows across cycles while maintaining a periodic Hubble parameter. Their principal finding is that despite the innovative approach to entropy management via a 'stiff' equation of state in contracting phases, such models remain geodesically incomplete, akin to traditional inflationary models.
Model Exploration
Cyclic cosmological models aim to offer a comprehensive framework that explains the observable universe without resorting to singularities. The IS model suggests circumventing the entropy problem through an exponentially growing scale factor ensuring an entropy dilution mechanism. Here, the Hubble parameter exhibits periodicity over time, whereas the scale factor's growth is exponential from one cycle to another, purportedly enabling the universe to extend infinitely in time.
Kinney and Stein analyze the IS model under the prism of geodesic completeness, utilizing the Borde-Guth-Vilenkin (BGV) theorem that historically demonstrated the past-incompleteness of inflationary spacetimes. Their approach evaluates whether the periodic growth of the scale factor, while ostensibly circumventing entropy accumulation, might introduce other issues like geodesic incompleteness.
Key Findings
A significant focus of the paper is on the implications of the IS model's periodic Hubble parameter, where the mean expansion rate, even when considering contractive phases, results in a net positive value across cycles. Such a condition leaves the spacetime geodesically past-incomplete due to the periodically occurring expansion that outweighs contraction. This is established by showing that for any spacetime with a net positive average Hubble parameter over a cycle, the past proper time is finite, making the spacetime incomplete. The continuity of the Hubble parameter and the scale factor across cycles, while furnishing a framework for entropy dilution, introduces a contradiction by restricting the spacetime from being truly infinite in the time dimension.
Implications and Future Directions
These findings carry profound implications for the theoretical foundations of cosmological models. While the IS cyclic model may present an elegant solution to the entropy problem, the trade-off with geodesic completeness necessitates re-evaluation. Models of this type prompt further research into possible modifications that might achieve both entropy management and geodesic completeness.
Moreover, these results signal the fundamental challenge in designing cyclic cosmological paradigms that simultaneously address entropy and spatial constraints without invoking the shortcomings demonstrated by existing inflationary models. Future directions could explore extensions or alternative formulations of cyclic models to balance the entropic and geodesic conditions more effectively.
In conclusion, "Cyclic Cosmology and Geodesic Completeness" provides a critical evaluation of bouncing cosmological models, highlighting intrinsic challenges around geodesic completeness. This scholarly contribution aids in refining theoretical cosmology and prompts further exploration of the nuanced interplay between cyclic dynamics and the foundational constraints dictated by relativity and thermodynamics.
