What can the observation of nonzero curvature tell us?

The eternally inflating multiverse provides a consistent framework to understand coincidences and fine-tuning in the universe. As such, it provides the possibility of finding another coincidence: if the amount of slow-roll inflation was only slightly more than the anthropic threshold, then spatial curvature might be measurable. We study this issue in detail, particularly focusing on the question: "If future observations reveal nonzero curvature, what can we conclude?" We find that whether an observable signal arises or not depends crucially on three issues: the cosmic history just before the observable inflation, the measure adopted to define probabilities, and the nature of the correlation between the tunneling and slow-roll parts of the potential. We find that if future measurements find positive curvature at \Omega_k < -10^-4, then the framework of the eternally inflating multiverse is excluded with high significance. If the measurements instead reveal negative curvature at \Omega_k > 10^-4, then we can conclude (1) diffusive (new or chaotic) eternal inflation did not occur in our immediate past; (2) our universe was born by a bubble nucleation; (3) the probability measure does not reward volume increase; and (4) the origin of the observed slow-roll inflation is an accidental feature of the potential, not due to a theoretical mechanism. Discovery of \Omega_k > 10^-4 would also give us nontrivial information about the correlation between tunneling and slow-roll; e.g. a strong correlation favoring large N would be excluded in certain measures. We also ask whether the current constraint on \Omega_k is consistent with multiverse expectations, finding that the answer is yes, except for certain cases. In the course of this work we were led to consider vacuum decay branching ratios, and found that it is more likely than one might guess that the decays are dominated by a single channel.