  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.
