That is, it presumes that if space is a hypersphere, then that hypersphere must sit in a four-dimensional Euclidean space, allowing us to view it from the outside. Nature, however, need not cling to this notion. It would be perfectly acceptable for the universe to be a hypersphere and not be embedded in any higher-dimensional space. It was Einstein himself, to prevent the universe from collapsing under its own gravity. He wanted the universe to be static, on average.
Trouble is, the energy density of vacuum fluctuations, naively computed, is infinite.
One can get rid of this infinity by assuming that fluctuations on a wavelength shorter than the Planck length are not allowed, but then the cosmological constant is still extremely large. Superstring theories predict that the energy density of vacuum fluctuations is zero, at least in a flat universe. This prediction is hailed as one of the successes of superstrings. When the cosmological term overcomes the effect of gravitational self-attraction, the universe expands at an exponentially increasing rate. The latest data on distant supernovas seem to imply that the expansion of the universe is accelerating, and therefore the cosmological constant is small but not zero.