Quantum cosmology of a dynamical Lambda

TL;DR

This paper explores the quantum cosmology of theories with a dynamical cosmological constant, showing that certain quantum states solve the Wheeler-DeWitt equation and suggesting implications for the probability distribution of Lambda.

Contribution

It demonstrates that the Chern-Simons state remains a solution in a class of theories with a dynamical Lambda, extending previous fixed Lambda results and analyzing parity branches.

Findings

The CS state solves the Wheeler-DeWitt equation with dynamical Lambda in the parity-even branch.

Without IR cutoff, the probability distribution for Lambda is a delta function at zero.

In the parity-odd branch, the CS state is not a solution; the most general solution allows for a range of Lambda predictions.

Abstract

By allowing torsion into the gravitational dynamics one can promote the cosmological constant, $\Lambda$, to a dynamical variable in a class of quasi-topological theories. In this paper we perform a mini-superspace quantization of these theories in the connection representation. If $\Lambda$ is kept fixed, the solution is a delta-normalizable version of the Chern-Simons (CS) state, which is the dual of the Hartle and Hawking and Vilenkin wave-functions. We find that the CS state solves the Wheeler-DeWitt equation also if $\Lambda$ is rendered dynamical by an Euler quasi-topological invariant, {\it in the parity-even branch of the theory}. In the absence of an infra-red (IR) cut-off, the CS state suggests the marginal probability $P(\Lambda)=\delta(\Lambda)$. Should there be an IR cutoff (for whatever reason) the probability is sharply peaked at the cut off. In the parity-odd branch, however, we can still find the CS state as a particular (but not most general) solution, but further work is needed to sharpen the predictions. For the theory based on the Pontryagin invariant (which only has a parity-odd branch) the CS wave function no longer is a solution to the constraints. We find the most general solution in this case, which again leaves room for a range of predictions for $\Lambda$.