How flat is the universe and what does flatness mean for geometry?

[Holocene]

Some questions:

How sure are we that the Universe is flat?

A flat Universe implies that there is precisely enough matter/energy in the universe to eventually halt the expansion?

What is really meant by "flat"? We theorize that space-time has 4 dimensions, so does "flat" simply mean that light rays are not distorted by the shape of the universe?

 

[hellfire]

How sure are we that the Universe is flat?

The observable universe is nearly flat, but the observations could be also consistent with some closed models on very large scales.

A flat Universe implies that there is precisely enough matter/energy in the universe to eventually halt the expansion?

A flat universe without dark energy implies that there is exactly the right amount of matter to have eternal expansion. If the matter density is greater the universe will be closed and will recollapse. This is only valid without dark energy. With dark energy there is no clear relation between fate of the universe and geometry of space; a closed model may also expand forever.

What is really meant by "flat"? We theorize that space-time has 4 dimensions, so does "flat" simply mean that light rays are not distorted by the shape of the universe?

Flat space is euclidean (three-dimensional) space in which the angles of a triangle sum 180° or in which initially parallel lines remain parallel forever. May be this thread will be interesting for you.