A Topological Extension of General Relativity to Explore the Nature of Quantum Space-time, Dark Energy and Inflation

Abstract

General Relativity is extended into the quantum domain. A thought experiment is explored to derive a specific topological build-up for Planckian space-time. The presented arguments are inspired by Feynman's path integral for superposition and Wheeler's quantum foam of Planck mass mini black holes/wormholes. Paths are fundamental and prime 3-manifolds like T^3, S^1xS^2 and S^3 are used to construct quantum space-time. A physical principle is formulated that causes observed paths to multiply: It takes one to know one. So topological fluctuations on the Planck scale take the form of multiple copies of any homeomorphically distinct path through quantum space-time. The discrete time equation of motion for this topological quantum gravity is derived by counting distinct paths globally. The equation of motion is solved to derive some properties of dark energy and inflation. The dark energy density depends linearly on the number of macroscopic black holes in the universe and is time dependent in a manner consistent with current astrophysical observations, having an effective equation of state w=-1.1 for redshifts smaller than unity. Inflation driven by mini black holes proceeds over n=55 e-foldings, without strong inhomogeneity, a scalar-to-tensor ratio r=ln(7)/n=0.036 and a spectral index n_s=1-r=0.964. A discrete time effect visible in the cosmic microwave background is suggested.