  The theory of eternal inflation in an inflaton potential with multiple vacua
predicts that our universe is one of many bubble universes nucleating and
growing inside an ever-expanding false vacuum. The collision of our bubble with
another could provide an important observational signature to test this
scenario. We develop and implement an algorithm for accurately computing the
cosmological observables arising from bubble collisions directly from the
Lagrangian of a single scalar field. We first simulate the collision spacetime
by solving Einstein's equations, starting from nucleation and ending at
reheating. Taking advantage of the collision's hyperbolic symmetry, simulations
are performed with a 1+1-dimensional fully relativistic code that uses adaptive
mesh refinement. We then calculate the comoving curvature perturbation in an
open Friedmann-Robertson-Walker universe, which is used to determine the
temperature anisotropies of the cosmic microwave background radiation. For a
fiducial Lagrangian, the anisotropies are well described by a power law in the
cosine of the angular distance from the center of the collision signature. For
a given form of the Lagrangian, the resulting observational predictions are
inherently statistical due to stochastic elements of the bubble nucleation
process. Further uncertainties arise due to our imperfect knowledge about
inflationary and pre-recombination physics. We characterize observational
predictions by computing the probability distributions over four
phenomenological parameters which capture these intrinsic and model
uncertainties. This represents the first fully-relativistic set of predictions
from an ensemble of scalar field models giving rise to eternal inflation,
yielding significant differences from previous non-relativistic approximations.
Thus, our results provide a basis for a rigorous confrontation of these
theories with cosmological data.
