Attractor solutions in scalar-field cosmology

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 $\ensuremath{\phi}$ and $\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}$ 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 ${m}^{2}{\ensuremath{\phi}}^{2}$ 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 $\ensuremath{\phi}\mathrm{\text{\ensuremath{-}}}\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\ensuremath{\phi}}$ variables and suggest a physical understanding of attractor behavior that is compatible with Liouville's theorem.