Attractor Solutions in Scalar-Field Cosmology

TL;DR

This paper examines the nature of attractor solutions in scalar-field cosmology, reconciling their apparent stability with Liouville's theorem by analyzing conserved measures in phase space.

Contribution

It demonstrates the existence of a conserved measure in certain scalar field models and explains attractor behavior through measure divergence, aligning with Hamiltonian dynamics.

Findings

Conserved measure exists for m^2 potentials.

Attractors correspond to divergences in the conserved measure.

Provides a measure-based interpretation of attractor behavior.

Abstract

Models of cosmological scalar fields often feature "attractor solutions" to which the system evolves for a wide range of initial conditions. There is some tension between this well-known fact and another well-known fact: Liouville's theorem forbids true attractor behavior in a Hamiltonian system. In universes with vanishing spatial curvature, the field variables (\phi,\dot\phi) specify the system completely, defining an effective phase space. We investigate whether one can define a unique conserved measure on this effective phase space, showing that it exists for m^2\phi^2 potentials and deriving conditions for its existence in more general theories. We show that apparent attractors are places where this conserved measure diverges in the (\phi,\dot\phi) variables and suggest a physical understanding of attractor behavior that is compatible with Liouville's theorem.