# Weyl curvature and Inflation of the universe

The idea of Roger Penrose is that in the first instants after the big bang, the Weyl tensor was close to zero. In his opinion, this could be a more reasonable physical explanation of the isotropy and homogeneity of the actual universe. Is this hypothesis still a valid alternative of the theory of Alan Guth and Alexei Starobinski on the inflationary universe?

I would not say that Penrose's Weyl curvature hypothesis is an alternative to inflationary cosmology, so much as it is something additional. Cosmology modeled as a de Sitter (dS) spacetime has no Weyl curvature. The dS spacetime has zero Weyl curvature, and in the Petrov classification is a type O solution to the Einstein field equations. The dS spacetime has none of the Killing symmetries that are eigenvalues of the Weyl curvature with Killing vectors. This means the cosmos as a dS spacetime, whether the universe as it is now with a small cosmological constant $\Lambda~\simeq~10^{-54}m^{-2}$ or with it was during inflationary period starting $10^{-35}s$ into the cosmos and lasting $10^{-32}s$ with $\Lambda~\simeq~10^{65}m^{-2}$, has a relatively small Weyl curvature over all.
Penrose's observation was the development of the universe has a growth in Weyl curvature as matter clumps. The Schwarzschild metric has zero Ricci curvature and in the vacuum a Weyl curvature that is responsible for the distortion of a spherical shell of test masses into an ellipsoid. This is also the tidal acceleration. This means the accumulation of matter into stars and other localized matter results in an increases in Weyl curvature. Even with black hole quantum evaporation the metric back reaction produces gravitational radiation that radiates Weyl curvature away. Weyl curvature get “dispersed” and transmitted to ${\cal I}^+$. With this part there is something of interest with how this Weyl curvature plays a role in quantum information.
During the inflationary period the accelerated expansion was enormous. Any local distribution of mass-energy, should it have existed, was rapidly inflated apart. This period $10^{-35}s$ to $10^{-32}s$ of the cosmos mostly likely enforced $C_{abcd}~=~0$. At the end the vacuum transitioned from this false high energy vacuum to a low energy vacuum we observe now. The energy gap then produced matter and radiation. With this emerged the “stuff” that can start of increase the Weyl curvature. It is also possible the vacuum transition had local variations, which would have set up small inhomogeneities and anisotropies that were “seeds” for later clumping.