  In the context of bubble universes produced by a first-order phase transition
with large nucleation rates compared to the inverse dynamical time scale of the
parent bubble, we extend the usual analysis to non-vacuum backgrounds. In
particular, we provide semi-analytic and numerical results for the modified
nucleation rate in FLRW backgrounds, as well as a parameter study of bubble
walls propagating into inhomogeneous (LTB) or FLRW spacetimes, both in the
thin-wall approximation. We show that in our model, matter in the background
often prevents bubbles from successful expansion and forces them to collapse.
For cases where they do expand, we give arguments why the effects on the
interior spacetime are small for a wide range of reasonable parameters and
discuss the limitations of the employed approximations.
