Transcript from Epic of Evolution: Life, the Earth and the Cosmos (BEP 210A)
April 7, 2000 - Lecture by Claude Bernard
First of all, I want to come back to the question Ursula asked me last time because I didn’t answer it well. The more I thought about it, the more I realized that there was a better answer than what I gave. Remember, I was explaining how --- in Einstein’s view --- the curvature of space is what makes objects move along curved paths when they are influenced by gravity. For example, in Einstein’s relativity the presence of the sun curves the space-time around it. And the picture I drew (Figure 1 --- which I’m repeating here from last lecture) makes an analogy with a two-dimensional space. So think of space as being a rubberized sheet, a two-dimensional sheet. The sun bends that sheet making a curved two-dimensional space. And then a planet moves along a geodesic (drawn in blue) in that curved space.
Figure 1: Curved
space around the sun, and a planet’s geodesic path
Now Ursula asked a good question about this picture. Basically she said, “Wait a minute, a planet orbits the sun: it follows a path with the sun at the center. If I look at your picture, the planet is going around a circle, but the sun is not at the center of the circle --- the sun is way below the center.”
To understand the picture, you have to keep in mind that it is an analogy in which we are pretending that space-time is two-dimensional. (There’s no way I could draw the real situation in which space-time is four-dimensional.) The creatures that live in the space of Figure 1 are two-dimensional creatures who can only move on the surface, on this rubberized sheet. Therefore, in their universe there is no direction “up-down.” On this curved two-dimensional space the center of this circle is the sun. If they move the same distance inward from any point in the orbit (always staying on the surface, which is their universe), they will come to the sun. So to them the center of their orbit is the sun.
We of course are three-dimensional creatures and can see, from outside their space, that the “true” center of the circle is a point “above” the sun. But that point is not in their universe and is not relevant to them.
Here’s a similar example: Suppose you got in a ship circled around Hawaii at a radius of 100 miles. Thinking of it in a two-dimensional way, you would say, “I’m going in a circle around Hawaii, since I’m always 100 miles away from it.” But if you really take into account that the Earth is a three-dimensional sphere, and look at the situation from the outside, then the center of the circle is actually some miles underneath Hawaii.
What reason do the two-dimensional creatures give for the circular motion of their planet? Their Newton would say, “The sun is pulling the planet towards the center of the circle, bending its path and preventing it from going off in a straight line.”
But their Einstein would say. “No, our space has been inherently curved by the sun. The planet is just moving along a geodesic in this curved space.”
Okay, I hope that helps to clear up the question Ursula raised.
Now as I said last time, Einstein’s picture explains why gravity affects all objects in the same way. Since each object just moves along the geodesic, all objects take exactly the same path. It won’t matter how big the object is or what it is made of. If you start it off in the same place with the same speed, it will take the same path. So if, for example, you took the planet in Figure 1 away and replaced it with a rock and sent the rock off with the same speed that the planet had, the rock would follow the same (blue) path.
In Newtonian physics, the fact that gravity affects all objects the same is a coincidence. There’s no fundamental understanding of it. It comes out of the fact that the motion of bigger objects is more difficult to change, but bigger objects are also heavier: gravity exerts a larger force on them. And the two effects just cancel each other, so every object --- independent of its mass --- ends up following the same path. But in General Relativity (Einstein’s theory of gravity), the reason each object takes the same path is because the path is just determined by the space. It has nothing to do with the object or with forces on the object. The object is just going along a geodesic and the geodesic, the shortest distance between two points, is the same for any object.
So General Relativity gives you a better understanding of this fundamental principle called the Principle of Equivalence, that all objects are affected by gravity in the same way. But it also immediately tells you that light will be bent by gravity. It’s not just planets and rocks and wads of paper that follow geodesics. Light will also follow a geodesic, and a geodesic is curved in a curved space-time.
We think of a light beam as our quintessential idea of a straight line. Take a flashlight and point it at something, the light beam makes a straight line. But that straight line is not straight, it’s a geodesic, and if the space is curved that “straight” line will itself be curved. Einstein realized this and predicted in 1915 that light coming to us from a distant star would be bent if it came near the sun. Since the sun bends space-time in its vicinity, any light passing near the sun should followed a curved path. Figure 2 shows what this would mean.
Figure 2:
Light from a distant star being bent by the sun
In Figure 2, the distant star is pink and the Earth is blue. Light (pink line) coming from the star follows a geodesic past the sun. Because the space is curved by the sun the geodesic curves. However, we when see the light arriving at the Earth, we assume it has been moving in the same direction the whole time. That is, we assume that the light has been following the brown line. Therefore, the apparent position of the star (i.e., the apparent source of the light) is the location shown in brown. Mark?
[Student: Is the light bent by the sun due to its gravity?]
Yes, Einstein would say the light is bent because it is traveling in a curved space-time, and the space-time has been curved by the sun. So a massive object curves the space-time around it, and then other objects moving in that curved space-time follow geodesics (the shortest distance between two points). But those geodesics will not appear straight in our usual sense. They will appear curved when viewed on a flat map or with our “flat space mentality.” It is the same reason that the shortest route between Los Angeles to Rome appears curved if you look on a flat map.
This provides another example of how science is “falsifiable.” Einstein new understanding of gravity allowed him to predict that light would be bent by the presence of a massive object. And this prediction could be tested. If light was bent, that would be evidence in favor of Einstein’s theory. If light was not bent or if it was bent by a different amount than what he claimed, that would be evidence against the theory. In the latter case, the theory would have had to be thrown out or revised.
Now Einstein’s prediction of the bending of light by the sun was made in 1915. But there was a difficulty in testing it right away. To test it, you need to observe light from a distant star passing near the sun and then determine how much that light is bent. But the problem is when a star is located in a direction close to the direction of the sun, the star is hard to see because the sun is so bright. In other words, you can’t see stars in the daytime, and especially not in the same part of the sky as the sun. Well, this problem has a solution: you wait for an eclipse.
There was actually an additional problem: 1915 was in the middle of World War I. So the test had to wait until the first suitable eclipse after the war. That eclipse was in 1919 in Africa and was observed by a British expedition. They pointed their telescopes in the direction of a star that happened --- because of the particular location of the Earth in its orbit at the time --- to be almost in the same direction from the Earth as the sun. The star appeared to be slightly displaced from its normal location. (The “normal location” of a star is its apparent position in the sky when its light does not have to pass near the sun to get to us.) In other words, the starlight was bent by the sun. In addition, the amount of bending was just as predicted by Einstein.
With this great success of his theory of gravity, Einstein became the most famous scientist ever. Before 1919, the general public hadn’t heard much about Einstein and hadn’t paid much attention. But at this point he burst on the international scene and became an object of public adulation. He remains almost a cult figure; Time magazine just named him the “person of the century.”
Of course, the 1919 eclipse was not the only case where light can be bent by the curvature of space-time. Whenever there’s light coming to us past some large intervening mass, the light will be bent. I’ll now show you some recent pictures.
[slide 1] What you can see here is an effect called “gravitational lensing.” When you look through an ordinary lens (for example, a camera lens) you can sometimes get an image that is spread out around a circle. (For camera buffs, think of the edges of a close-up taken with a wide-angle lens.) And that’s what is happening to the light from a distant galaxy in this picture. But the light is not being bent by an ordinary lens; it is being bent by the gravitational effect of an intervening cluster of galaxies. The light from the distant galaxy ends up spread out into these rounded arc-like shapes. You can see several of them from different distant galaxies being bent by the intermediate cluster. The process is known as gravitational lensing. Of course, unlike in a camera lens, here a “lens designer” does not control the direction and amount of the bending of light. The bending is randomly determined by the shape and amount of the intermediate mass --- so it’s not going to focus it and make a beautiful picture of the distant galaxy. It’s going to be an out-of-focus lens in general, as we see in this slide.
[slide 2] Here’s another example. In this case, the light gets bent in various ways, and it actually appears to us to be coming from different places. Imagine a some light rays from the star in Figure 2 passing on the other side of the sun --- closer to the front of the picture – and being bent towards the Earth. Then we would side light coming to us from two directions, which would give a double image of the distant star. That is what is happening here. In fact, there are about four or five different images of the same galaxy. It just appears to be in various places because the light from it gets bent and takes various paths to us.
[slide 3] This one shows two more dramatic examples of multiple images. In each case light from a distant galaxy passes an intermediate galactic cluster. The light is bent and reaches us by various paths, so that we see four different images of the distant galaxy. Of course these are very far away and the resolution isn’t very good. When we turn the room lights back on, I’ll talk a more about that so you can understand such multiple images better.
[slide 4]. This is the furthest example of gravitational lensing that we have. These blobs form a cluster of galaxies at an intermediate distance, a mere two or three billion light-years away. And this crescent of light here is a very distant galaxy, about 13 billion light-years away. (The light has taken 13 billion years to get to us.) Since the Big Bang was only about 14 billion years ago, this distant galaxy is about as far as away as you can get. Its light has been bent to make it appear to have the crescent shape. Using General Relativity, we can determine the amount of bending. When that bending is removed mathematically, the actual galaxy is found to have a shape much more like the picture at the bottom --- much less like a crescent and more like a sphere.
Now let me just say a couple words more about the multiple images you saw. It’s just one galaxy behind another one. The light from the distant one comes out in various directions (up, down, left, right) and it’s bent around the intermediate galaxy to us from each of those directions. So, for example, we can see four images in certain cases.
[Ursula: Why is it four?]
I don’t actually know why the case of four images seems to be so common. Of course, it depends on the shape of the intervening matter, and you can get various numbers of images. But I’ve never gone through the calculations to see why the four-image case tends to occur often.
[Ursula: I guess there’s no reason why a galaxy should be round.]
That’s true. If the intervening galaxy were perfectly round and the distant galaxy were directly behind it, then you would get an image that was one complete circle around the intervening galaxy. (Slide 1 was almost like that.) But in usual cases, the intervening matter is not particular round and you get all kinds of funny sets of images. And in fact the intervening matter is often not one galaxy but a cluster of galaxies. Of course all that says it that the image won’t be a single complete circle. But I don’t know why it is that four images are so common.
So that’s one amazing consequence of Einstein’s General Relativity: the bending of light due to gravity. Another prediction that Einstein’s theory makes is the existence of black holes. As I’ve said before, a black hole is a region where the matter is so dense that nothing can get out --- not even light. Therefore it appears black.
In Newton’s theory a black hole is not possible. The more matter you have in a star, the greater the gravitational effect it would have (the harder it would pull on objects) and the harder it would be for something to escape from the star. But in Newtonian physics, no matter how massive and dense a star was, you could always escape if you had enough energy. There would never be any situation where it would be impossible; it would just be difficult. The more matter and the denser the matter, the more difficult it would be. But there’s a big difference between difficult and impossible.
In Einstein’s theory, once the matter gets dense enough nothing can get out. We can understand that result in a rough way by the pictures I’ve been drawing of the two-dimensional analogues of bent space. The explanation will not be perfect, and it will not be possible to understand black holes in all their glory, but I think you will get some idea of the difference between viewing gravity as a force (the Newtonian picture) and viewing it as the curvature of space-time.
We start off with Figure 3, which is basically the same picture as Figure 1. A concentration of mass (shown in red) is warping the space-time around it. As I’ve said many times, the presence of matter curves space-time.
Figure 3:
Matter bending space-time
Now imagine adding more mass to the central concentration. The space-time is curved more drastically, as in Figure 4.
Figure 4: More
mass; more curvature
Now as more and more mass is added, eventually you can have the situation where the curvature has gotten so great that a piece of space-time separates off from the rest, as shown in Figure 5.
Figure 5: Black hole formed.
The separate piece of space-time is a black hole. Since it’s now become a separate space, there’s no way that anything in it can ever get out to the old space. Any light inside would just go round and round the black hole space. So an observer in the old space would not see any light coming from the black hole.
The difference with the Newtonian view is that here we’re talking about curved space, and you can imagine a piece of space completely separated off. That’s not something that can happen when we just think of gravity as a force --- because bigger and bigger forces can always be overcome by bigger and bigger amounts of energy. And so a black hole does not exist in Newton’s theory but it does exist in Einstein’s theory.
Now, as I’ve said, this analogy isn’t perfect: If you push it, there will be features of black holes that this picture doesn’t get right. But I think it does give you some intuitive feeling for what a black hole is. The main problem with such pictures is that the real curvature is not of space but of space-time, and there’s no way I can draw a curved space-time on the blackboard. Time is not something that you can easily picture as a dimension, so that’s mainly where the analogy fails. But the picture shows some of the features of the real situation. Yes?
[Student: Is that place where it breaks off the event horizon?]
Yeah, that’s what we would call the “event horizon”: the place where the upper space now stops because the black hole has broken off. But to be completely honest, let me say that the event horizon is one of the features that are imperfectly described by this picture. Other questions?
I want to describe one more feature of black holes. Again, it’s something that distinguishes Einstein’s theory from Newton’s. Remember how we made black holes. They are formed when a big star collapses. The star first reaches the neutron star stage where all the neutrons are pushed right up against each other. But if the star has too much mass, the pressure from the neutrons will not be enough to hold it up --- the gravitational effect is so strong that the collapse continues.
Now you might imagine that there could be some new kind of force that at that point --- a force that we haven’t yet discovered --- that could create enough pressure to stop the star from collapsing into a black hole. The idea would be that you could create some new kind of exotic star analogous to a white dwarf or a neutron star, but with some new force holding it up. Is that a possible? The answer is no. In General Relativity, gravity or curvature of space is created not just by the presence of mass or energy (mass and energy are of course equivalent) but also by the presence of pressure. That feature of General Relativity is just totally different from anything in Newtonian physics.
So what matters in determining the strength of gravity in a collapsing star is not just the total amount of mass or energy but also the amount of pressure. It is then possible to show that if there were some new type of force which created some enormous pressure (in an effort to stop the collapse into a black hole), then the gravitational effect created by the new pressure would overwhelm the pressure and make star collapse all the faster. And my General Relativity teacher in graduate school called this effect “gravitational judo.” It uses the strength of the hypothetical new force against itself. So the greater the strength of the new force in trying to hold up the star, the more curvature of space-time will be created and the more it will collapse.
The result is that there’s no way that even as-yet-unknown forces could halt the collapse to a black hole. We think it will actually stop collapsing --- this is now an area of speculation --- only when it compresses to such very, very small distances that the laws of General Relativity itself are changed. To stop the collapse, gravity must have some very different behavior in some realm that hasn’t been investigated yet. This change would probably have to do with the quantum mechanical effects on General Relativity. But these are things of which we know very little, if anything.
Okay, that’s all I want to say about black holes.
Another amazing thing about General Relativity is that you can apply it as a theory of gravity to the universe as a whole. Alexander Friedmann, a Russian physicist, first used General Relativity in this way. (There’s some discussion of Friedmann’s interesting life in the Ferris book ‑‑- he had a hard life and died at an early age, but contributed a great deal to physics.) In 1922 Friedmann was able to show under certain mild assumptions that there are only three possible shapes for the entire universe.
Friedmann’s assumptions are the ones I mentioned when I first talked about the Big Bang: the universe “homogeneous” and “isotropic.” Homogeneous means that the universe is the same everywhere, if you look at it on a big enough scale. If you put out a big enough “net” you’re going to catch the same variety of “fish” wherever you fish in the universe. With a big enough net you’ll always catch approximately the same number of spiral galaxies, the same number of elliptical galaxies, the same amount of empty stuff between them, the same amount of dust and gas, and so forth. That seems to be true anywhere in the universe. (Of course, on a smaller scale the universe is not homogeneous. For example if your net is just big enough to hold a single galaxy then in some places you’ll catch a spiral one; in some places, an elliptical one; and in many places, none at all.)
Isotropic means that the universe looks the same in every direction. I won’t go into the explanation of why that is not implied by the assumption of homogeneity --- there are subtle differences. Friedmann actually had to assume separately that the universe (on large enough scales) is homogeneous and isotropic.
Now with these assumptions, Friedmann analyzed how the matter and energy in the universe could curve the space-time of the universe. He showed using the equations of General Relativity that there are only three types of possibilities for the way the universe could be shaped by the matter and energy in it.
The three possibilities are shown on transparency #11. First let me emphasize that these are pictures of two-dimensional spaces, whereas the real universe has three spatial dimensions plus time. But since we can’t draw a curved three- or four-dimensional space, we draw these two-dimensional analogues of the real thing.
Possibility “A” in the transparency is a spherical shape, what’s called a “closed geometry” in this context. When I talked about the Big Bang early in the course, I represented it by a balloon, which has a spherical shape. (Again, a balloon is a two-dimensional surface, and is therefore only an analogy to the actual space.) Now you can see that the balloon-type shape for the universe is actually only one possibility.
The universe could also have what’s called an open geometry (possibility “B”), which is a kind of a saddle shape. Let me emphasize that in the open geometry case, the universe doesn’t end at the edge of the picture shown, but continues out in all directions infinitely. That is why it’s called “open” --- there is no end to the space. A closed geometry like a sphere is a finite amount of space. So it’s possible that our universe is a finite amount of space like possibility A, but it’s also possible that it’s an infinite amount of space like in possibility B. (When Auden says, in the poem assigned for Discussion Section, “… but who / Would feel at home astraddle / An ever expanding saddle?” he’s talking about possibility “B”.)
The third possibility (“C”) is, in some sense, in between the other two. It is possible that our universe is flat, that the net gravitational effect of all the mass and energy turns out to make a flat universe, which is curved in neither of the ways A or B. This one is also infinite but (in the two-dimensional analogy) like a flat sheet extending in all directions.
In all three of these cases, the space would expand as time goes on. That is the expansion of the universe that we’ve talked about many times. If the first possibility is the right one, the universe is like a balloon that is being blown up. But in the other two cases, the universe also expands. In these two cases space would be like an infinite rubberized sheet (either flat [C] or curved [B]) being stretched out in all directions as time goes on. Mark?
[Student: If it’s flat like in the case of flat geometry, how can it expanding in all directions? How does it do that without becoming a sphere?]
Remember these pictures are two-dimensional analogues of the real situation. The real situation for case C would be an infinite three-dimensional space, which is being stretched equally in all three directions at once. It can do that and still stay “flat” --- in the sense that geodesics remain straight lines and the angles of triangles still add up to 180 degrees. The analogy in the two-dimensional case (which we can picture much better) would be a flat two-dimensional rubberized sheet being stretched equally in both directions (in the plane) but staying flat as that happens.
You may also be bothered the question of how a sheet, which is already infinite, can still stretch. Of course an infinite sheet can’t get any “bigger,” but the sheet can still stretch in the sense that the distance between any two points in it is increasing with time.
So you can have expansion in any of these three examples. Of course the one I used before was a balloon because that is the easiest to picture, and because it’s not possible to blow up a saddle or a flat sheet like it is to blow up a balloon.
Let me make one other point. In describing the Big Bang, I started with a beach ball sized object. What was that beach ball? It was the size of our current observable universe at a time shortly (10-32 seconds) after the Big Bang, the time that I began my story in these lectures. That the beach ball sized region has now expanded into our current observable universe. In other words, light emitted from the edges of that region at 10-32 seconds after the Big Bang is now, 14 billion years later, just arriving at our location. That light has traveled a total of 14 billion light years in distance to get to us. [It started just a foot away at 10-32 seconds, but the space in between has been expanding so rapidly that it’s only getting to us now!]
Now there is nothing “special” about the observable universe: there is no “brick wall” at the edge of it now and there was no brick wall at the edge of it at 10-32 seconds after the Big Bang. The observable universe is just that --- the part of the entire universe that happens to be observable by us right now. The part of the universe observable by a hypothetical intelligent being in a galaxy 10 billion light years away is a different region, which only overlaps partly with our observable universe. Similarly, in a billion years, our observable universe will include stuff further way, which is not observable right now. So in a billion years, a physicist talking about the observable universe at the time 10-32 seconds after the Big Bang would have to discuss a region which is somewhat bigger than a beach ball.
So there is no reason at all to think that the “beach ball” at time 10-32 seconds was the entire universe. On the contrary, we have a good (but not iron-clad) reason to suspect that the beach ball was only a very tiny part of the entire universe. [I hope to discuss this reasoning in my last lecture.] Now the two-dimensional analogue of a beach ball is a circle. So I’ve drawn little circles to represent the beach ball on each of the possibilities in transparency #11. So even if possibility A is the right one and the entire universe is shaped like a sphere, that sphere is not the beach ball. The beach ball is just some tiny region of the whole sphere. Our observable universe is only a miniscule part of the entire universe.
Recall that in the very first lecture I said that ultimately I would have to be careful about what I meant by “the universe.” And now you can see one reason to be careful. Our observable universe is only what’s inside this little circle, but the entire universe could be something much, much bigger.
[Just to stretch your brains, let me remark that the term “entire universe” could also have various interpretations. For example suppose the universe had the shape of possibility A --- the sphere. But what if there was another sphere “somewhere else?” (The location of the other sphere’s center would have to be some point separated by a distance in the “fifth dimension” from the center of our sphere.) Would that other sphere be part of the “entire universe?” Would your answer depend on whether there was some way, in principle, if not in practice, for the matter in one sphere to affect the matter in the other?].
[Ursula: Are you going to tell us which of the three possibilities is right?]
No. I can’t tell you, because we don’t know anything about the shape of the entire universe.
We do know a lot about the geometry of our observable universe, that little patch. But knowing about the geometry of the little patch is not --- as you will see --- the same as knowing about the geometry of the entire universe. In fact, as far as we know now, understanding the geometry of the entire universe is irrelevant for understanding the geometry of our little patch. So as far as the shape of the entire universe I could say: a) we don’t know, and b) we don’t care, except in an emotional or philosophical sense. It could be, though, that when we understand physics a lot better than we do now, the shape of the entire universe may become important in a physical sense.
[Student: What about our little patch? Is it round?]
Our patch is flat, as close as we can tell. In fact that was a great puzzle: Why should everything be adjusted so that our patch would be flat? As I just explained, the flat situation is the borderline between two other situations, and it seemed unlikely that we should have landed right on the borderline. [I’ll explain this more my final lecture.]
I need to explain more about what determines the geometry of the entire universe, or of our observable universe. The amount of curvature of a space depends on the (average) density of energy in the space. (We could also talk about the density of mass, but that’s the same thing since energy and mass are equivalent.) So the density of energy in the universe determines which of the three possibilities is the one that’s actually realized.
Now there is a certain special density of energy --- which we can calculate --- called the “critical density.” If the actual density of energy in the universe is greater than the critical density, the space will curve into a ball or sphere. So in that case we’ll get possibility A, a closed or sphere-like universe. Remember that the presence of mass (or energy) bends space. If there’s a lot of mass, then space is bent so much that it curves up into a ball.
On the other hand, for density less than the critical amount we will have a situation like B, an open geometry. There’s not enough mass and energy to curve space back on itself, it’s sort of curving “out” in this case.
And it’s only for energy density exactly equal to the critical amount that we have the borderline case, which is the flat geometry.
To repeat: Density less than critical density gives an open geometry, the saddle. Density greater than critical density gives a closed geometry, the balloon. And density equal to the critical density gives the flat situation. Saying it this way makes the flat situation look incredibly unlikely because it would require a density exactly equal to the critical amount. It’s the “Goldilocks” situation: to get a flat universe, the density has to be neither too small nor too big --it has to be just right. And that would seem to be very unlikely.
In fact, when astronomers began to realize that our observable universe did appear to be flat or very close to it, that issue was a nagging problem. It was called the “Flatness Problem.” Why should our density be so close (or exactly equal) to the critical density? If the density was random then the chance of hitting it right on the money is 0, and the chance of getting very close would be very small. It would like trying to guess a number between 1 and 10, but where the number didn’t have to be a whole number. For example, imagine how hard it would be to guess the number 3.987432649235746492837498255347212124. You would be either above or below it, but not right on.
So that’s the so-called Flatness Problem. We actually have some understanding --- although it’s somewhat speculative --- of how the observable universe came to be so flat. I hope to talk about that in my last lecture.
As I’ve said, we think that we’re very close to the flat situation at least in our little observable patch. Our little patch doesn’t appear to be curved in any way. How do we know what the shape of our observable patch is? The answer is something I can explain, and it involves the Cosmic Microwave Background (CMB). Remember that the CMB consists of the photons left over from the Big Bang. Those photons separated out from matter at the period called decoupling (about 300,000 years after the Big Bang), when atoms formed for the first time and photons no longer had significant interactions with matter. A photon from the CMB that hits a detector on Earth today has very probably been going in a “straight line” (along a geodesic) since the period of decoupling. The photon just has very little chance of hitting something that would deflect it from its geodesic between then and now.
Now, to a very good approximation, the energy (temperature) of CMB photons is the same not matter which part of the sky they arrive from. This is an indication that the universe at the time of decoupling was quite homogeneous --- a more or less uniform distribution of matter. However, in recent years, very accurate measurements of the CMB show that there are some small fluctuations: regions of the sky from which arriving photons are very slightly cooler or warmer than usual (i.e., have very slightly less or more energy than usual). What does that imply? It tells us that at the time of decoupling there was some slight lumpiness in the universe. There would be regions where the matter was slightly denser or slightly less dense than the surrounding regions. Today, this lumpiness is quite extreme: there are galaxies, where matter is clumped together quite densely, and then spaces between galaxies, which are almost empty.
The CMB tells us that the lumpiness was very small at the time of decoupling --- only about one part out of 10,000. Why would the lumpiness affect the microwave background radiation? There are many reasons; perhaps the simplest to understand is that when photons were leaving a region with slightly more than the average density, they would have experienced a backward gravitational pull slightly greater than normal. In consequence they would have lost a little more than the usual amount of energy --- that means they would arrive at Earth a little more red shifted than the typical photon. Similarly, a photon coming from a region with slightly less than the average density of matter would have been pulled back slightly less than average and would arrive at Earth slightly less red shifted than the typical photon.
We actually can deduce something about the size of the clumps. Initially, immediately after the Big Bang, there were probably random fluctuations of all sizes. But clumps would then start to grow because of gravity --- wherever there happened to be more matter than average, it would have greater gravitational attraction than average, and pull in additional surrounding matter, becoming denser. But there is a limitation here. Nothing, not even gravitational attraction, can move faster than light. In the 300,000 years between the Big Bang and decoupling, gravitational attraction could only travel a distance of 300,000 light-years. Therefore, clumps of size larger than 300,000 light years across would not have had enough time to increase their density significantly. Really dense clumps could be at most 300,000 light years in diameter.
Now suppose we survey the microwave background radiation coming to us from every direction. We look for regions in the sky from which the microwave photons coming to us have more than the average red shift. Among various such regions, we choose the largest ones. The size of those regions must have been about 300,000 light years across when the photons were emitted.
Furthermore, the radius of the have observable universe when the light was emitted was about 15 million light years. We thus have a triangle like that shown in Figure 6.
Figure 6: A triangle
formed from a region of higher than usual red shift in the cosmic microwave
background
In Figure 6, the red region is a clump in the matter at the time of decoupling. The purple lines are the paths of cosmic microwave photons toward the center of the observable universe (i.e. our current position --- although of course the Earth was not there yet). We know the length of every side of this triangle. Now in the meantime, of course, the universe has expanded. All sides of this triangle have grown. But since they all grow simultaneously, the shape of the triangle would not have changed. In a flat space, the angles of a triangle are completely determined once we know the lengths of all the sides. So if we measure the angle between the two purple lines in Figure 6, we can compare that angle to what it would be in flat space. If the angle is greater than the flat space angle, we know that our observable universe curved like a sphere (possibility A). If the angle is less than the flat space angle, we know that our observable universe curved like a saddle (possibility B). And if the angle is the same as the flat space angle, we know that our observable universe is flat (possibility C).
When the experiment was actually done (within the last two years!), it was found that the angle was just what it should be in a flat space (about 1 degree). So our observable universe seems to be flat. I will discuss the reasons for that is in my last lecture.
[Note: I have just tried to give a basic understanding of how we determine the shape of the observable universe; I have omitted most of the subtleties. For example, we don’t actually look at just one triangle. Since we are talking about effects due to random fluctuations of the density in the early universe, we must average over many such triangles over the entire sky. Similarly, my claim that the smallest lump is 300,000 light years across should really be modified to something like, “The typical small size lump is 300,000 light years across.”]
[end of lecture]