R grew rapidly from an unimaginably hot, dense state that existed immediately following this singularity (when R had a small, finite value); this is the essence of the Big Bang model of the universe. Understanding the singularity of the Big Bang likely requires a quantum theory of gravity, which has not yet been formulated. Third, the curvature index k determines the sign of the mean spatial curvature of spacetime averaged over sufficiently large length scales (greater than about a billion light-years). If k=1, the curvature is positive and the universe has a finite volume. A universe with positive curvature is often visualized as a three-dimensional sphere embedded in a four-dimensional space. Conversely, if k is zero or negative, the universe has an infinite volume. It may seem counter-intuitive that an infinite and yet infinitely dense universe could be created in a single instant when R = 0, but exactly that is predicted mathematically when k does not equal 1. By analogy, an infinite plane has zero curvature but infinite area, whereas an infinite cylinder is finite in one direction and a torus is finite in both. A toroidal universe could behave like a normal universe with periodic boundary conditions. The ultimate fate of the universe is still unknown because it depends critically on the curvature index k and the cosmological constant Λ. If the universe were sufficiently dense, k would equal +1, meaning that its average curvature throughout is positive and the universe will eventually recollapse in a Big Crunch, possibly starting a new universe in a Big Bounce. Conversely, if the universe were insufficiently dense, k would equal 0 or −1 and the universe would expand forever, cooling off and eventually reaching the Big Freeze and the heat death of the universe. Modern data suggests that the rate of expansion of the universe is not decreasing, as originally expected, but increasing; if this continues indefinitely, the universe may eventually reach a Big Rip. Observationally, the universe appears to be flat (k = 0), with an overall density that is very close to the critical value between recollapse and eternal expansion. Multiverse hypotheses Some speculative theories have proposed that our universe is but one of a set of disconnected universes, collectively denoted as the multiverse, challenging or enhancing more limited definitions of the universe. Scientific multiverse models are distinct from concepts such as alternate planes of consciousness and simulated reality.