geometry of the universe

On a flat tabletop, the angles of any triangle add up to exactly 180 degrees, and two parallel lines never meet. But draw a triangle on the curved surface of a globe and its angles add up to more than 180 degrees; the surface is curved. The same question can be asked of the whole universe: is the large-scale geometry of space flat, like the tabletop, or curved? The geometry of the universe is the science of which of three possibilities space obeys.

Einstein's gravity says that the amount of matter and energy in the universe determines the curvature of space, and it allows three overall shapes. If the density equals the critical density (Omega = 1), space is flat: parallel light rays stay parallel and cosmic triangles have angles summing to 180 degrees. If density exceeds critical (Omega greater than 1), space is closed, curving back on itself like the surface of a sphere — finite in size with no edge, and triangles bulge past 180 degrees. If density is below critical (Omega less than 1), space is open, curving like a saddle, with triangles falling short of 180 degrees. These are three-dimensional analogues of two-dimensional surfaces, so they are hard to picture, but the triangle test is exactly how the curvature is measured in practice.

Geometry matters because it ties the universe's contents to its shape, and historically to its fate. The measurement comes mainly from the cosmic microwave background: hot and cold spots in it have a known true size, and how big they appear on the sky depends on whether the light travelled through flat or curved space, like reading curvature from how a distant object's apparent size is distorted. The verdict is striking: the universe is flat to within about half a percent. An honest note: flat is the measured result, but it is flat to within our precision — we cannot prove it is perfectly flat, only that any curvature is very small.