If the density of the Universe is greater than critical density, it’s closed. Think of a sphere which’ll stop expanding, then collapse.
If the density of the Universe is less than critical density, it’s open, continuing to expand forever at a slowing rate.
If the density of the Universe is equal to critical density, it’s flat or open since it’ll keep expanding but at a slower and slower rate until the rate is zero.This looks to be best option so far since dark energy which appears to exert a repulsive force larger than gravity is pushing the Universe apart and is three times more abundant than matter. Even the matter component is probably mostly dark (brown dwarfs and so on, other stuff we can’t for some reason see).
What is critical density? We can calculate it. Think of an expanding spherical dust cloud. An object of mass m moves away from the centre of the cloud, mass M with a speed v that satisfies Hubble’s Law v=Hr where H is Hubble’s Constant, currently thought to be 72km/s/Mpc
Replacing v with Hr and M with density x volume:
If the energy is positive (the kinetic energy term larger ) the mass will continue to move away. The mass will stop moving at infinity if the energy terms add up to zero and contract back if the energy is negative. It is the critical density term which thus defines what happens:
On a graph then, even though its density = critical density, it doesn’t actually look as if the size of the Universe is going to flatten out as time goes on any time soon, instead, it’ll get bigger more rapidly than before because the expansion rate is accelerating.