This page is designed to illustrate the relationship between the energy density and curvature of the universe.
This frame depicts a two dimensional universe in three dimensional space. It models our own three-dimensional universe thought to exist inside a higher dimensional space.
Although the universe bends into the third dimension, it remains a 2D surface. People travelling through it wouldn't notice moving in the third dimension. Sorf of like how when you move along the surface of the Earth, you don't notice the Earth's curvature!
The shape of the universe is determined by how much stuff (in the form of energy) is in it. We describe the amount of stuff by (Omega), known as the density parameter. More stuff (energy) means a larger value for , and less stuff means a smaller value. The exact relationship is given by something called the Friedmann equations.
is defined as the ratio between the energy density () of the universe and the critical energy density, , where is the exact amount of energy necessary to make the universe flat.
Thus, for the universe is flat!
When , the universe is said to have positive curvature, or be closed. This is because in the absence of dark energy, this kind of universe will eventually collapse down on itself
When , the universe is said to have negative curvature, or be open. This kind of universe will expand forever even without dark energy.
Experiment with the evolution of the universe here
So what is the value of in our universe?
According to the most recent (2015) results of the Planck Survey, is within 0.005 of 1. So we live in a flat universe!