Classical and quantum trace-free Einstein cosmology

TL;DR

This paper explores classical and quantum solutions of trace-free Einstein cosmology using conformal time, revealing cyclic behavior in closed universes and a discrete spectrum for the cosmological constant, with implications for different matter fields.

Contribution

It provides a solvable model of trace-free Einstein cosmology in classical and quantum regimes, analyzing the spectrum of the cosmological constant across different universe geometries.

Findings

Closed universe exhibits cyclic evolution with finite scalar curvature.

Quantum spectrum of the cosmological constant is discrete and positive for closed universe.

Flat and open universes have continuous spectra for the cosmological constant.

Abstract

Trace-free Einstein gravity, in the absence of matter fields and using the Friedmann-Robertson-Walker (FRW) metric, is solvable both classically and quantum mechanically. This is achieved by using the conformal time as the time variable and the negative or positive of the inverse of the scale factor as configuration variable to write the classical equation of motion, which turns out to be the one of a free particle ($k=0$), a harmonic oscillator ($k=1$), and a repulsive oscillator ($k=-1$) in the real half-line. In all cases, the observable identified as the cosmological constant is six times the Hamiltonian. In particular, for a closed Universe ($k=1$), spacetime exhibits a cyclic evolution along which the scalar curvature is constant and finite, thereby avoiding singularities. The quantum theory is reached by using canonical quantization. We calculate the spectrum of the observable corresponding to the cosmological constant. Remarkably, for the closed Universe ($k=1$), the spectrum is discrete and positive while for flat ($k=0$) and open ($k=-1$) universes, the spectra are continuous. Heisenberg's uncertainty principle imposes limitations on the simultaneous measurement of the Hubble expansion (momentum variable) and the configuration variable. We also report the observable identified as the cosmological constant for inflaton, phantom and perfect fluids coupled to trace-free Einstein gravity in the FRW metric.