Kantowski–Sachs cosmology, Weyl geometry, and asymptotic safety in quantum gravity

Abstract

A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvement of the Schwarzschild black hole that removes the r = 0 singularity is presented. It is followed with a RG improvement of the Kantowski–Sachs metric associated with a Schwarzschild black hole interior such that there is no singularity at t = 0 due to the running Newtonian coupling G(t) vanishing at t = 0. Two temporal horizons at [Formula: see text] and [Formula: see text] are found. For times below the Planck scale t < t P, and above the Hubble time t > t H, the components of the Kantowski–Sachs metric exhibit a key sign change, so the roles of the spatial z and temporal t coordinates are exchanged, and one recovers a repulsive inflationary de Sitter-like core around z = 0, and a Schwarzschild-like metric in the exterior region z > R H = 2G o M. The inclusion of a running cosmological constant Λ(t) follows. We proceed with the study of a dilaton-gravity (scalar–tensor theory) system within the context of Weyl’s geometry that permits singling out the expression for the classical potential [Formula: see text], instead of being introduced by hand, and find a family of metric solutions that are conformally equivalent to the (anti) de Sitter metric. To conclude, an ansatz for the truncated effective average action of ordinary dilaton gravity in Riemannian geometry is introduced, and a RG-improved cosmology based on the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric is explored where instead of recurring to the cutoff identification k = k(t) = ξH(t), based on the Hubble function H(t), with ξ a positive constant, one now has [Formula: see text], when [Formula: see text] is a positive-definite dilaton scalar field that is monotonically decreasing with time.