Taming singularities and chaos in conformal gravity

TL;DR

This work addresses cosmological singularities within Weyl-invariant conformal gravity, focusing on the Bianchi IX universe. By employing conformal transformations with a conformal factor $\Omega(x)$ and a dilaton $\phi$, the authors show that singularities are gauge artifacts in the conformal phase and identify an analytic subclass of frames that are geodesically complete for massless, conformally coupled, and massive probes, while recasting dynamics into a quasi-FLRW form to tame chaos. They construct explicit rescalings, e.g. $S(t) = 1 + \frac{L^{2n}}{t^{2n}}$, that cause the affine-time to reach $t=0$ or $t_f$ to diverge, removing both the big bang and big crunch; analogous results hold for conformally coupled and massive particles. Additionally, a conformal rescaling with $S(t) = 1 + (t/t_0)^k$ can suppress Mixmaster chaos by equalizing Kasner exponents, steering the near-singularity dynamics toward isotropy. The findings show that singularities and chaotic behavior in Bianchi IX can be resolved within a purely gravitational, Weyl-invariant framework, with potential implications for early-universe cosmology and the role of conformal symmetry in geodesic completeness.

Abstract

We hereby address the cosmological singularity problem in a general gravitational theory invariant under Weyl conformal transformations. In particular, we focus on the Bianchi IX spacetime and we show that both the initial (big bang) and final (big crunch) singularities disappear in an infinite class of conformal frames naturally selected according to analyticity. It turns out that the past and future singularities are both unattainable within a finite affine parameter (for massless particles) or within a finite proper time (for massive and conformally coupled particles). In order to prove such a statement, we show the geodesic completion of the spacetime when probed by massless, massive, and conformally coupled particles. Finally, the chaotic behavior of the spacetime near the singularity is tamed by a conformal rescaling that turns the Bianchi IX metric into a quasi-FLRW spacetime.