Final answer:
An open universe, with hyperbolic geometry and density less than critical, expands forever, while a closed universe, with spherical geometry and density higher than critical, eventually collapses.
Present evidence suggests our universe is flat, with density equal to the critical density, aligning with Euclidean geometry and supporting infinite expansion that slows indefinitely.
Step-by-step explanation:
Distinguishing Between a Closed and an Open Universe
The nature of the universe, in terms of its geometry and fate, is determined by its density relative to the critical density. An open universe is one that will expand forever and has a hyperbolic geometry where two initially parallel light rays will diverge. The density of matter in an open universe is less than the critical density. On the other hand, a closed universe will eventually stop expanding, potentially collapsing back on itself, and has a spherical geometry where parallel light rays will converge. A closed universe has a density higher than the critical density.
Current evidence, such as measurements in the universe's expansion rate and observations of the Cosmic Microwave Background Radiation (CMBR), supports the idea that our universe is flat; meaning, its density is equal to the critical density. In this scenario, parallel light rays never meet and the universe will not recollapse but expansion slows to a halt infinitely far in the future. This flat universe theory aligns with Euclidean geometry. The density and composition of dark matter also play a role in the universe's fate, with neutrino mass and various forms of leptonic matter contributing factors.