Canonical measure and the flatness of a FRW universe

TL;DR

This paper examines the measure problem in FRW universes with scalar fields, analyzing how different measures impact the flatness problem and the role of quantum cosmology in resolving ambiguities.

Contribution

It clarifies the ambiguity in the measure for flatness in FRW universes with bounded potentials and discusses how quantum cosmology can resolve this ambiguity.

Findings

The measure ambiguity persists for bounded potentials.

Quantum cosmology input can resolve measure ambiguities.

The measure aligns with classical signature change approaches.

Abstract

We consider the claim of Hawking and Page that the canonical measure applied to a FRW universe with a massive scalar field can solve the flatness problem regardless of inflation. By considering a general potential we are able to understand how the ambiguity for $\Omega$ found by Page in the $R^2$ theory is present when the potential is bounded from above. We suggests reasons why such potentials are realistic. The ambiguity for the possibility of inflation present in the Gibbons- Hawking- Stewart measure can be resolved by an input from Quantum cosmology. We contrast the resulting measure obtained in this way with those obtained in the more usual approaches to quantum cosmology. The measure agrees with that found from a "classical" signature change.