Phenomenological parameterization of quintessence

We propose a simple phenomenological parameterization of quint-essence with a time-varying equation of state. In particular, it accounts for the possibility of early dark energy. The quintessence potential can be reconstructed in terms of the present fraction in dark energy, the present equation of state and the amount of early dark energy. Distinguishing quintessence-a time varying dark energy component-from a cosmological constant is a major quest of present observational cos-mology. The first particle physics models [1, 2] are based on a scalar field rolling down a potential. Typically, a cosmic attractor solution [1, 2] renders the evolution of the scalar field independent of the initial conditions (after a certain transition time in early cosmology). For a given scalar potential the quintessence cosmology is as predictive as a cosmological constant scenario. However, there are numerous models with different potentials. For cosmolog-ical tests of their general features one would like a simple parameterization of the cosmological dynamics, say the Hubble parameter as a function of redshift, H(z), that can later be translated into statements about the form of the quintessence potential. A first obviously important parameter is the present fraction in homogeneous dark energy [1] Ω (0) h = Ω h (z = 0). For observations at low z the next important parameter is the first derivative of Ω h (z) or the present equation of state w 0 = w h (z = 0) [3]. These two quantities are directly related, cf. eq. (3) below. Continuing an expansion in z is not very meaningful, however, if one wants to cover the physics at high z, like last scattering. For example , the CMB-anisotropies are very sensitive to the fraction of dark energy at last scattering [4] Ω (ls) h = Ω h (z = 1100) and structure formation depends crucially on the weighted mean of Ω h during structure formation ¯ Ω (sf) h [5, 6].