  We develop a set of controlled, analytic approximations to study the effects
of bubble collisions on cosmology. We expand the initial perturbation to the
inflaton field caused by the collision in a general power series, and determine
its time evolution during inflation in terms of the coefficients in the
expansion. In models where the observer's bubble undergoes sufficient slow-roll
inflation to solve the flatness problem, in the thin wall limit only one
coefficient in the expansion is relevant to observational cosmology, allowing
nearly model-independent predictions. We discuss two approaches to determining
the initial perturbation to the inflaton and the implications for the sign of
the effect (a hot or cold spot on the Cosmic Microwave Background temperature
map). Lastly, we analyze the effects of collisions with thick-wall bubbles,
i.e. away from the thin-wall limit.
