Recent observations suggest that a large fraction of the energy density of the universe has negative pressure. One explanation is vacuum energy density; another is quintessence in the form of a scalar field slowly evolving down a potential. In either case, a key problem is to explain why the energy density nearly coincides with the matter density today. The densities decrease at different rates as the universe expands, so coincidence today appears to require that their ratio be set to a specific, infinitessimal value in the early universe. In this paper, we introduce the notion of a "tracker field," a form of quintessence, and show how it may explain the coincidence, adding new motivation for the quintessence scenario.A number of recent observations suggest that Ω m , the ratio of the (baryonic plus dark) matter density to the critical density, is significantly less than unity.1 Either the universe is open, or there is some additional energy density ρ sufficient to reach Ω total = 1, as predicted by inflation. Measurements of the cosmic microwave background, the mass power spectrum, 1-3 and, most explicitly, the luminosity-red shift relation observed for Type Ia supernovae, 4 all suggest that the missing energy should possess negative pressure (p) and equationof-state (w ≡ p/ρ). One candidate for the missing energy is vacuum energy density or cosmological constant, Λ for which w = −1. The resulting cosmological model, ΛCDM, consists of a mixture of vacuum energy and cold dark matter. Another possibility is QCDM cosmologies based on a mixture of cold dark matter and quintessence (−1 < w ≤ 0), a slowly-varying, spatially inhomogeneous component.7 An example of quintessence is the energy associated with a scalar field (Q) slowly evolving down its potential V (Q).5-8 Slow evolution is needed to obtain negative pressure, p = 1 2Q 2 − V (Q), so that the kinetic energy density is less than the potential energy density.Two difficulties arise from all of these scenarios. The first is the fine-tuning problem: Why is the missing energy density today so small compared to typical particle physics scales? If Ω m ∼ 0.3 today the missing energy density is of order 10 −47 GeV 4 , which appears to require the introduction of new mass scale 14 or so orders of magnitude smaller than the electroweak scale. A second difficulty is the "cosmic coincidence" problem:9 Since the missing energy density and the matter density decrease at different rates as the universe expands, it appears that their ratio must be set to a specific, infinitessimal value in the very early universe in order for the two densities to nearly coincide today, some 15 billion years later.What seems most ideal is a model in which the energy density in the Q-component is comparable to the radiation density (to within a few order of magnitude) at the end of inflation, say. If there were some rough equipartition of energy following reheating among several thousands of degrees of freedom, one might expect the energy density of the Q-component to be two or so orders of ma...

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?

Parametric resonance has been discussed as a mechanism for copious particle production following inflation. Here we present a simple and intuitive calculational method for estimating the efficiency of parametric amplification as a function of parameters. This is important for determining whether resonant amplification plays an important role in the reheating process. We find that significant amplification occurs only for a limited range of couplings and interactions. ͓S0556-2821͑98͒05604-5͔ PACS number͑s͒: 98.80.Cq

Recently, we introduced the notion of "tracker fields," a form of quintessence which has an attractor-like solution. Using this concept, we showed how to construct models in which the ratio of quintessence to matter densities today is independent of initial conditions. Here we apply the same idea to the standard cold dark matter component in cases where it is composed of oscillating fields. Combining these ideas, we can construct a model in which quintessence, cold dark matter, and ordinary matter all contribute comparable amounts to the total energy density today irrespective of initial conditions.