A classification of scalar field potentials with cosmological scaling solutions

TL;DR

The paper analyzes a flat FRW cosmology with a perfect fluid and a scalar field, asking which scalar-field potentials yield scaling solutions where $\rho_\phi$ scales as a power of the scale factor, $\rho_\phi \propto R^{-n}$, while the background fluid scales as $\rho \propto R^{-m}$ with $m=3\gamma$. It derives three classes of potentials that realize this behavior: the well-studied exponential (tracker, $m=n$), negative power-law potentials ($m>n$), and a newly identified positive power-law class ($m<n$), providing exact solutions and stability analyses for each. The authors perform a phase-plane analysis, find attractor conditions, and show that negative power-laws yield attractors for all $\alpha$, while positive power-laws require $\alpha>2\left(\frac{6+m}{6-m}\right)$ for attractor behavior, with era-dependent thresholds. They also discuss applications to late-time acceleration, including the ZWS potential, which can mimic tracker behavior early and evolve toward a sub-dominant scalar density later, highlighting practical tuning considerations. Overall, the work offers a complete classification of scaling potentials and clarifies their dynamical properties and cosmological implications.

Abstract

An attractive method of obtaining an effective cosmological constant at the present epoch is through the potential energy of a scalar field. Considering models with a perfect fluid and a scalar field, we classify all potentials for which the scalar field energy density scales as a power-law of the scale factor when the perfect fluid density dominates. There are three possibilities. The first two are well known; the much-investigated exponential potentials have the scalar field mimicking the evolution of the perfect fluid, while for negative power-laws, introduced by Ratra and Peebles, the scalar field density grows relative to that of the fluid. The third possibility is a new one, where the potential is a positive power-law and the scalar field energy density decays relative to the perfect fluid. We provide a complete analysis of exact solutions and their stability properties, and investigate a range of possible cosmological applications.

Figures (2)

• Figure 1: Three phase planes for the radiation-dominated $m=4$ case. From top to bottom, $\alpha = 8$, $10$, $12$. The top panel spirals outwards, and the bottom one inwards.
• Figure 2: The evolution of the energy densities in a radiation-dominated Universe with $\alpha = 30$. The vertical axis is in arbitrary units. The dotted line is the radiation energy density, and the solid line is the scalar field potential energy. The scalar field kinetic energy, shown as the dashed line, oscillates through zero out of phase with the potential energy, but does so too sharply for the plotting resolution.