How Do We Know the Universe is Flat?

Of all the forces at work in nature, gravity is at the same time the simplest and most
enigmatic of all.
It is simple because it is easily created and, at least on the scale of the everyday,
easily described. All that is needed to create it is mass. If you have mass, you have gravity.
It is enigmatic because we really don't know much more about it than that. We know that
it takes mass to make it, but simple questions like, 'Why does gravity only pull?', "Why
is gravity so weak compared to other forces?" are difficult to understand.
It is, at once, simple and complicated.
Gravity is the force that sculpts creation, and through it, the universe takes its form.
Under its influence, interstellar gas collects to form clouds that, under the inexorable
crush, are ignited into stars. These stars are collected into galaxies, and galaxies
are gathered into structured tendrils, reaching throughout the cosmos.
The effects of gravity are everywhere, and this force, more than any other in nature,
is responsible for the shape of all that is.
So, what is the form of the universe? How can we find out?
The main characteristic that gives the universe its shape is density. The density of the universe,
the amount and location of all matter, define where the gravity of the cosmos is concentrated,
which bends and warps spacetime as described by Einstein's General Theory of Relativity.
In order to uncover the geometry of the universe, we need to be able to see all of it, at one
time. Fortunately we have such pictures. This is an image of the entire universe in the
microwave region of the spectrum. This is what the universe looked like when it was
only 380 thousand years old.
This is the radiant heat left over from the big bang.
This image is full of fluctuations, tiny variations in temperature. The red parts are slightly
warmer than the blue parts by only on part in a thousand, but they are measurable and
it is these fluctuations that give us the information we need to determine the shape
of the universe.
It turns out, these tiny changes are related to fluctuations in the density of matter in
the universe and thus carry information about the initial conditions for the formation of
cosmic structures such as galaxies, clusters of galaxies, and voids.
To understand this relationship, all we need is basic geometry of triangles. On a flat
surface, all angles in a triangle add up to 180 degrees and parallel lines remain parallel.
On a positively curved surface, like a sphere, they add up to something greater than 180
and parallel lines converge. A surface with negative curvature, such as a saddle, the
three angles add up to less than 180 degrees and parallel lines diverge.
So, to measure the shape of the universe, all we need is a triangle, a really big one,
one that covers the entire universe, and measure the angles.
The all sky map of the CMB provide us with just such a surface within which to measure
our triangle, with the Earth at one apex and two more points on the image, we can make
our measurement.
The fluctuations in the CMB are randomly placed spots with an apparent size of about 1 degree
across. They are produced by sound waves that travel through the hot ionized gas in the
universe at a known speed (the speed of light divided by the square root of 3) for a known
length of time (380,000 years).
Knowing the rate and time, we can obtain the distance to what is known as the last scattering
surface - the remnants of a cosmic cloudbank. Simplifying a little, this distance, along
with the Hubble constant and the actual light path taken by the CMB to our eyes, will tell
us the geometry.
If the universe is flat, our triangle would have straight lines and all angles would equal
180 degrees and the average angular distance betwen CMB fluctuations would be 1 degree.
If the universe was positively curved, our lines bend outward and our angles would be
greater than 180 degrees, and the angular distance would be about one and a half degrees
across. A negatively curved universe would look like this with an average angular distance
of a half a degree.
After careful measurements of the CMB using WMAP data, the average distance between fluctuations
was found to be 1 degree with an accuracy of 15% and a error of 2%.
The universe, it appears, is flat.
Currently the Planck Space Telescope is taking a higher resolution map of the Cosmic Microwave
background, and very soon, we hope to have an even more accurate answer to the question
of the shape of existence.