Future supernovae data and quintessence models

The possibility of unambiguously determining the equation of state of the cosmic dark energy with existing and future supernovae data is investigated. We consider four evolution laws for this equation of state corresponding to four quintessential models, i.e. (i) a cosmological constant, (ii) a general barotropic fluid, (iii) a perfect fluid with a linear equation of state and (iv) a more physical model based on a pseudo-Nambu-Goldstone boson field. We explicitly show the degeneracies present not only within each model but also between the different models: they are caused by the multi-integral relation between the equation of state of dark energy and the luminosity distance. Present supernova observations are analysed using a standard chi(2) method and the minimal chi(2) values obtained for each model are compared. We confirm the difficulty in discriminating between these models using present type Ia supernovae data only. By means of simulations, we then show that future SNAP observations will not remove all the degeneracies. For example, wrong estimations of Omega(m) with a good value of chi(2) (min) could be found if the right cosmological model is not used to fit the data. Finally, we give some probabilities for obtaining unambiguous results, free from degeneracies. In particular, the probability of confusing a cosmological constant with a true barotropic fluid with an equation of state different from -1 is shown to be 95 per cent at a 2sigma level.