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.Comment: 46 pages, 5 figures; reference updates and typo corrections arising
from final Phys. Rev. D copy editin
