A proposal first advanced by Stephen Hawking and Jim Hartle, the no-boundary universe is one in which the universe does not start with a singularity. It uses American physicist Richard Feynman’s proposal to treat quantum mechanics as a “sum over histories,” meaning that a particle does not have one history in space-time but instead follows every possible path to reach its current state. By summing these histories—a difficult process that must be done by treating time as imaginary—you can find the probability that the particle passes through a particular point.
Hawking and Hartle then wedded this idea to general relativity’s view that gravity is just a consequence of curved space-time. Under classical general relativity, the universe either has to be infinitely old or had to have started at a singularity. But Hawking and Hartle’s proposal raises a third possibility—that the universe is finite but had no initial singularity to produce a boundary (thus the name).
The geometry of the no-boundary universe would be similar to the geometry of the surface of a sphere, except it would have four dimensions instead of two. You can travel completely around Earth’s surface, for instance, without ever running into an edge. In this analogy, unfolding in imaginary time, Earth’s North Pole represents the Big Bang, marking the start of the universe. (But just as the North Pole is not a singularity, neither is the Big Bang).