Stochastic Quintessence
  The behavior of the quintessence field is studied during inflation. In order to have a satisfactory model of dark energy, the quintessence field value today should be as insensible to the initial conditions as possible. Usually, only the dependence on the initial conditions specified at the end of inflation or, equivalently, at the beginning of the radiation dominated era, is considered. Provided the quintessence field is initially within a large but, crucially, finite interval, its present value becomes independent of the initial value it started from. The question as to whether inflation naturally drives the quintessence field to the above-mentioned interval is addressed. Since the quantum effects turn out to be important, the formalism of stochastic inflation is used in order to calculate the evolution of the quintessence field. Moreover, the quantum effects originating from the inflaton field are also taken into account and are proved to be sub-dominant in most cases. Finally, the requirement that the quintessence field is on tracks today is shown to imply quite tight constraints on the initial values of the quintessence and inflaton fields at the beginning of inflation. In particular, the initial value of the inflaton field cannot be too large which indicates that the quintessential scenario seems to be compatible with inflation only if the total number of e-folds is quite small. 
Jerome Martin and Marcello A. Musso