Cosmological Tracking Solutions

TL;DR

This work shows that quintessence fields with tracker behavior can robustly solve the coincidence problem by converging to a common evolutionary track across a vast range of initial conditions, eliminating fine-tuning of the early energy budget. It derives the Tracker Equation and the central condition $\Gamma=V''V/(V')^2>1$ (nearly constant) that guarantees tracking with $w_Q< w_B$ for broad potentials, including inverse-power laws and mixed-term forms like $V(Q)=M^{4+\alpha}/Q^{\alpha}$ and $V(Q)=M^4\exp(1/Q)$. The authors quantify the resulting $\Omega_Q$–$w_Q$ relation, show how $\Omega_Q$ grows to drive late-time acceleration, and discuss practical constraints (initial conditions, stability, and borderline/hybrid scenarios). Crucially, tracker models predict a non-negligible departure of $w_Q$ from $-1$ for realistic $\Omega_Q$, offering testable differences from a cosmological constant via CMB and supernova observations. Overall, the tracker framework provides a broad, relatively parameter-light path to explain the recent dominance of dark energy without fine-tuning, while yielding distinctive observational signatures.

Abstract

A substantial fraction of the energy density of the universe may consist of quintessence in the form of a slowly-rolling scalar field. Since the energy density of the scalar field generally decreases more slowly than the matter energy density, it appears that the ratio of the two densities must be set to a special, infinitesimal value in the early universe in order to have the two densities nearly coincide today. Recently, we introduced the notion of tracker fields to avoid this initial conditions problem. In the paper, we address the following questions: What is the general condition to have tracker fields? What is the relation between the matter energy density and the equation-of-state of the universe imposed by tracker solutions? And, can tracker solutions explain why quintessence is becoming important today rather than during the early universe?

Figures (9)

• Figure 1: Energy density versus red shift for the evolution of a tracker field. For computational convenience, $z=10^{12}$ (rather than inflation) has been arbitrarily chosen as the initial time. The white bar on left represents the range of initial $\rho_Q$ what leads to undershoot and the grey bar represents overshoot, combining for a span of more than 100 orders of magnitude if we extrapolate back to inflation. The solid black circle represents the unique initial condition required if the missing energy is vacuum energy density. The solid thick curve represents an "overshoot" in which $\rho_Q$ begins from a value greater than the tracker solution value, decreases rapidly and freezes, and eventually joins the tracker solution.
• Figure 2: A plot of $w_Q$ vs. red shift for the overshoot solution shown in Figure 1. $w_Q$ rushes immediately towards $+1$ and $Q$ becomes kinetic energy dominated. The field freezes and $w_Q$ rushes towards $-1$. Finally, when $Q$ rejoins the tracker solution, $w_Q$ increases, briefly oscillates and settles into the tracker value.
• Figure 3: A plot of $\dot{x}/6 =(1/6)\, d \, \ln{x}/d\, \ln{a}$ for the overshoot solution shown in Figures 1 and 2. At late times when $Q$ settles into the tracker solution, $\dot{x}$ is small and $w_Q$ is nearly constant. During the overshoot phase, $\dot{x}$ undergoes large positive and negative changes, as described in the text.
• Figure 4: The convergence of different initial conditions to the tracker solution. As derived in the text, $w_Q$ decays exponentially fast to the tracker solution combined with small oscillations. All the curves are for $V(Q)= M^4/Q^6$. The solid curve is the overshoot case from Figure 1. The thin dashed curve with $w \approx 0$ is the tracker solution which is overlaid for most $z$ by the dash-dotted curve, which represents a slightly undershooting solution.
• Figure 5: A comparison of the overshoot for three different models beginning from $\Omega_i = 10^{-3}$ (equipartition). The thick solid line is for $V(Q)= M^4/Q^6$, the thick dash-dotted line is for $V=M^4[exp(1/Q)-1]$, and the thick long dashed line is for $V(Q)= M^4/Q$. In all three examples, $Q$ falls rapidly downhill and freezes. In the first and second examples, $Q$ begins to roll again and joins the tracker solution before matter-radiation equality; the third example, which violates the condition derived in the text, does not begin to roll again by the present epoch.