We discuss fits of cosmological dark energy models to the available data on high-redshift supernovae. We consider a conventional model with Cold Dark Matter and a cosmological constant (LambdaCDM), a model invoking super-horizon perturbations (SHCDM) and models based on Liouville strings in which dark energy is provided by a rolling dilaton field (Q-cosmology). We show that a complete treatment of Q-cosmology requires a careful discussion of non-equilibrium situations (off-shell effects). The two main high-redshift supernova data sets give compatible constraints on LambdaCDM and the other models. We recover the well-known result that LambdaCDM fits very well the combined supernova data sets, as does the super-horizon model. We discuss the model-dependent off-shell corrections to the Q-cosmology model that are relevant to the supernova data, and show that this model fits the data equally well. This analysis could be extended to other aspects of cosmological phenomenology, in particular to the CMB and Baryon Acoustic Oscillations, which have so far been treated using on-shell models.
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