Perturbation evolution in cosmologies with a decaying cosmological constant

Structure formation models with a cosmological constant are successful in explaining large-scale structure data, but are threatened by the magnitude-redshift relation for type Ia supernovae. This has led to discussion of models where the cosmological ``constant'' decays with time, which might anyway be better motivated in a particle physics context. The simplest such models are based on scalar fields, and general covariance demands that a time-evolving scalar field also supports spatial perturbations. We consider the effect of such perturbations on the growth of adiabatic energy density perturbations in a cold dark matter component. We study two types of model, one based on an exponential potential for the scalar field and the other on a pseudo Nambu Goldstone boson. For each potential, we study two different scenarios, one where the scalar field presently behaves as a decaying cosmological constant and one where it behaves as dust. The initial scalar field perturbations are fixed by the adiabatic condition, as expected from the inflationary cosmology, though in fact we show that the choice of initial condition is of little importance. Calculations are carried out in both the zero-shear (conformal Newtonian) and uniform-curvature gauges. We find that both potentials allow models which can provide a successful alternative to cosmological constant models.