Weyl Curvature Hypothesis in light of Quantum Backreaction at Cosmological Singularities or Bounces

Penrose's 1979 Weyl curvature hypothesis (WCH) \cite{WCH} assumes that the universe began at a very low gravitational entropy state, corresponding to zero Weyl curvature, namely, the FLRW universe. This is a simple assumption with far-reaching implications. In classical general relativity the most general cosmological solutions of the Einstein equation are that of the BKL-Misner inhomogeneous mixmaster types. How could WCH and BKL-M co-exist? An answer was provided in the 80s with the consideration of quantum field processes such as vacuum particle creation, which was copious at the Planck time $10^-43 sec$, and their backreaction effects were shown to be so powerful as to rapidly damp away the irregularities in the geometry. It was proposed that the vacuum viscosity due to particle creation can act as an efficient transducer of gravitational entropy (large for BKL-M) to matter entropy, keeping the universe at that very early time in a state commensurate with the WCH. In this essay I expand the scope of that inquiry to a broader range, asking how the WCH would fare with various cosmological theories, from classical to semiclassical to quantum, focusing on their predictions near the cosmological singularities (past and future) or avoidance thereof, allowing the Universe to encounter different scenarios, such as undergoing a phase transition or a bounce. We point out that regardless of what other processes may be present near the beginning and the end states of the universe, the backreaction effects of quantum field processes probably serve as the best guarantor of WCH because these vacuum processes are ubiquitous, powerful and efficient in dissipating the irregularities to effectively nudge the Universe to a near-zero Weyl curvature condition.