Consistent Cosmic Bubble Embeddings

The Raychaudhuri equation for null rays is a powerful tool for finding consistent embeddings of cosmological bubbles into a background spacetime in a way that is largely independent of the matter content. We find that spatially flat or positively curved thin wall bubbles surrounded by a cosmological background must have a Hubble expansion that is either contracting or expanding slower than the background, which are more stringent constraints than those obtained by the usual Israel thin-wall formalism. Similarly, a cosmological bubble surrounded by Schwarzschild space, occasionally used as a simple "swiss cheese" model of inhomogenities in an expanding universe, must be contracting (for spatially flat and positively curved bubbles) and bounded in size by the apparent horizon.