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# Shape of the Universe

So far in this chapter we have stuck to quite an old fashioned view of the universe. We assumed that objects simply move freely through space, pulling on each other with gravity in proportion to their mass. In 1916 Albert Einstein published his theory of general relativity. In it, he declared that gravity was not a force as we know it, but rather a warping of space and time. For example, planets in orbit follow the contours of curved space-time, where the curve is created by the mass of the Sun.

Earlier in this chapter, the force of gravity was central in determining whether the universe is open, closed or flat. If general relativity is true, and we cannot explain the fate of the universe by a gravitational force, then whether the universe is open, closed or flat becomes a question of geometry. We will develop this idea in the second half of this quiz.

Imagine you are walking across a flat desert. This desert is your world. You cannot fly, and you cannot dig, so you effectively live in a two-dimensional world. You see what looks like an oasis in the distance.

What will be the shortest route to follow for you to get there?

When you reach the oasis it disappears—a mirage!

However, you now see before you a hill. Your world is not perfectly flat after all. You are sure that behind the hill will be a better, greener land.

Which will be the shortest route to the other side of the hill?

In the desert, you were moving in a two-dimensional plane, but this plane was bent by the hill into a third dimension: height. Spacetime curvature in general relativity is analogous to the hill.

Although the space we inhabit is three-dimensional, Einstein's insight was that it bends, as if into another dimension that we cannot observe. We often try to show this by drawing space as if it were two-dimensional, so that we can use the third dimension to show how space can bend. Massive objects like Earth or the Sun warp space like they are distorting a stretchy sheet.

In Einstein's model, why does the moon follow a circular path?

The universe experienced a big bang at its inception; the cosmic microwave background is provides a wealth of evidence supporting this theory.

The universe is called closed if the force of gravity eventually pulls all the matter in the universe all back together again. In our new paradigm, there is no force of gravity—only objects moving in curved space.

Which sketch below could show a universe where objects moving directly away from each other would eventually come back together again?

The three pictures from the previous question are how cosmologists view the three possible fates of the universe that we have considered—closed, open and flat—and this explains why they are named in this way.

The analysis we have done in terms of Newtonian gravity is still valid, but realizing that the fate of the universe is intimately tied to the shape of the universe both lends the question greater significance and gives us another way to attack it. We have tried to measure the mass density of the universe, and found that even with dark matter it is much less than the critical density, so we should be living in an open universe (the saddle-shaped one).

In this quiz we will see there a way we can measure the shape of the universe and learn its fate.

Imagine you have marked out a triangle on a flat piece of ground like shown. If you were to go the the three points of the triangle, measure the angles, and add them up, what value would you get to (in degrees)?

Now imagine you do the same thing again, but this time on a much bigger scale. You put point A of your triangle in Ecuador, on the equator but $$78^\circ$$ west. You place point B in Uganda, still on the equator but $$32^\circ$$ East. You place point C on the North pole.

Now add up the angles at the three points of the triangle. What is the total?

We have seen that on a (closed) curved surface like the Earth, the angles of a triangle add up to more than $$180^\circ$$. If the universe were open (the saddle shape), the angles of a triangle would add up to less than $$180^\circ$$. Essentially we can find out the shape of the universe by drawing a big triangle and measuring the angles in it.

Imagine we take two of the most distant objects we can find in our universe and draw a triangle joining them to the Earth. Thinking back to what you have learned in earlier chapters, which side are you unable to measure directly?

For any two objects, we can also measure the angular size of the distance between them. Suppose the two objects in the previous question are separated by $$1^\circ.$$

If we knew that the universe was flat, having two sides and an angle would be enough information to find the size of the other sides and angles. But we don't know it's flat—that's what we're trying to find out!

Suppose the side length $$d=\SI{1e7}{ly}$$ and we have strong reasons to believe the distance between the two objects is $$x=\SI{2e5}{ly}.$$ Would this provide evidence for an open universe, closed universe or flat universe?

Because the three angles of a triangle only sum to $$180^\circ$$ on a flat surface, forming a triangle from our position and two visible objects in the universe can reveal the universe's large-scale curvature. This approach to assessing the shape of the universe has two drawbacks:

1. The scale of the triangle must be similar to the scale on which the universe is homogeneous.
2. The lengths of all three sides must be known.

The second point is somewhat more difficult to overcome than the first, but in the next quiz, we will go through one way to overcome it.

We have seen so far that the ultimate fate of the universe is likely to be determined by its mass density, and under Einstein's (highly successful and tested) theory of general relativity, this also tells us the universe's shape. Measurements of the universe's mass density have thrown up mysterious "dark matter", and also suggested that the amount of mass in the universe is only about $$30\%$$ of the critical density needed for a flat universe, suggesting our universe is probably open. Until the late 1990s, this was as much as was known. Then along came two startling discoveries that we will reveal in the last quiz of this chapter.

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