(My latest book *God vs. Darwin: The War Between Evolution and Creationism in the Classroom* has just been released and is now available through the usual outlets. You can order it from Amazon, Barnes and Noble, the publishers Rowman & Littlefield, and also through your local bookstores. For more on the book, see here. You can also listen to the podcast of the interview on WCPN 90.3 about the book.)

For previous posts in this series, see here.

In addition to the appearance of dark matter, another interesting development arose when observers tried to determine the curvature of the universe, an important fact in determining the ultimate fate of the universe.

To understand this consider, as an analogy, a ball thrown upwards from the surface of the planet. It will slow down as it goes up due to the gravitational attraction of the planet’s mass. But will the ball eventually fall back to the ground or will it escape from the planet and go on forever? The answer depends on both the speed of the ball and the size of the planet. For a given speed of the thrown ball, if the mass of the planet is below a certain value, its gravitational pull on the ball is not sufficient to bring it back and the ball will escape and travel out in space forever.

The same feature holds for the universe. We currently know the speeds of the galaxies as they move apart form each other. We know that the gravitational field of the other galaxies is trying to slow them down. Whether the expansion eventually stops and the universe starts collapsing again or whether the expansion of the universe goes on forever depends of the combined mass of all the other galaxies, or more precisely, the density of the universe. And in turn, the density of the universe determines the shape of the universe.

If the density of the matter in the universe is below a certain value that we can calculate (called the ‘critical density’), the standard Big Bang theory predicts that the universe curves at every point in the shape of a saddle (called negative curvature) and will expand forever.

If the density of the universe is greater than the critical density, theory predicts that the universe curves the opposite way like a sphere (called positive curvature) and will stop expanding at some point and then start to collapse back into itself, like a thrown ball falling back to Earth.

Thus the ultimate fate of the universe is dependent on the curvature of the universe, which in turn is directly related to whether the actual density is greater or less than the critical density. The ratio of the actual density to the critical density is given by the Greek letter Ω and if this quantity is greater than 1, the universe is said to be closed (finite), if it is less than one it is said to be open (infinite and saddle shaped), and if it is exactly equal to one, it is said to be flat (and infinite). This figure from a NASA website provides a visualization by analogy with 2D space.

So clearly, knowing the curvature of the universe would give us important information about the ultimate fate of the universe. There are two ways to do this: measuring the density of the universe, calculating Ω, and thus inferring the curvature as above, or by directly measuring the curvature itself. Measurements of all the *visible* matter in the universe seems to indicate that the density of the universe is well below the critical density, signaling a saddle shape, and that we will have perpetual expansion. Even adding in all the postulated dark matter still gives a density that is only about 20-40% of the critical density.

But it is also possible to directly measure the curvature of space. How does one directly measure the curvature of space while living within that space? An analogy with the Earth may help. We currently live on the surface of the Earth. People have known for more than two thousand years that the Earth was a sphere. For most of that time, they inferred it indirectly, by observing eclipses, ships sinking over horizon, and so forth. In more recent times people have had direct confirmation for its spherical shape as a result of having circumnavigated the globe and viewed the Earth from outer space.

But it is theoretically possible for someone to determine the curvature of the Earth even if they never leave their living room or look outside, provided they have very precise measuring instruments. All they would have to do is draw a triangle on a sheet of paper that is laid flat on the ground (as shown in the figure), measure the three angles, and add them up. As all students are told, the total should be 180 degrees. But what many don’t know is that this result is a very special case that only occurs if the sheet of paper is flat.

If the surface of the Earth is curved into a sphere (and the sheet of paper follows that curvature), the sum of the angles will be greater than 180 degrees. You can easily see that this is true by imagining that we could draw a triangle large enough that one of its vertices is the North Pole and the other two vertices are on the Equator. We see that the two angles formed at the equator are each 90 degrees, which means that the sum of the three angles must be greater than 180 degrees. If the surface of the Earth had been saddle-shaped, the sum would be less than 180 degrees. The sum of the angles of a triangle drawn on a small sheet of paper would differ from 180 degrees by only a tiny amount, which is why you need precision instruments to measure the curvature of the Earth’s surface this way.

To directly measure the curvature of space in an analogous manner, a satellite called the Wilkinson Microwave Anisotropy Probe (WMAP) was launched in 2001 and the surprising result that it returned (with an astoundingly low 2% margin of error) was that the universe is neither saddle shaped nor spherical but flat, which meant that Ω=1 and hence the density of the universe must be almost exactly equal to the critical density. The unlikely coincidence of the actual density being equal to the critical density cries out for an explanation.

The ‘inflationary model’ of the universe, which is an add-on to the standard Big Bang theory, says that the very early universe underwent an extraordinarily rapid expansion within a tiny fraction of the very first second of life of the universe. This theory has gained widespread acceptance because a ‘flat’ universe would be an outcome, in addition to also solving what is known as the ‘horizon’ problem, which I will not go into.

So assuming that the universe is indeed flat, what is the source that is making the density of the universe exactly equal to the critical density? The solution that has been proposed is that space is filled with something called ‘dark energy’ that fills the entire universe (dark matter is assumed to only be present in galaxies) and this provides the amount of energy needed to make the universe flat.

But what is this new form of energy? And where did it come from?

Next: The cosmological constant and dark energy.

**POST SCRIPT: Crazy health care opponents**

I have not been writing recently about the health care issue even though it is important because a lot of recent activities was pure theater, mainly posturing and parliamentary maneuvering. But I will get back to it after the Big Bang series ends.

But what amazed me watching the process unfold me was the irrational and over-the-top rhetoric that was being thrown around by reform opponents. This video clip of the people at the demonstration last weekend gives a taste of the ignorance and selfishness prominently on display.

## 2 comments

## Paul Jarc

March 24, 2010 at 12:42 pm (UTC -4) Link to this comment

In today’s entry, you talk about the curvature of space through time, while in the What lies beyond the edge of the universe? entry, it sounds like you’re talking about the curvature of space at one moment in time. Are they necessarily the same? I can imagine that space could possibly have spherical curvature, making it finite at any one time, but it still might expand forever, making it ultimately infinite.

## Mano

March 24, 2010 at 4:18 pm (UTC -4) Link to this comment

Paul,

If the space has positive curvature, it means that it will stop expanding and start to collapse so it would not become infinite.