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# ASTR 160: Frontiers and Controversies in Astrophysics

## Lecture 17

## - Hubble's Law and the Big Bang (cont.)

### Overview

Class begins with a review of magnitudes and the problem set involving magnitude equations. Implications of the Hubble Law and Hubble Diagram are discussed. Professor Bailyn elaborates on the Big Bang theory of cosmology and addresses controversial questions related to the age, development, and boundaries of the universe. The fate of the universe, and possibly its end (known as the Big Crunch) are addressed. Imagining an expanding three-dimensional universe is proposed. The lecture ends with a question-and-answer session during which students inquire about a variety of topics related to cosmology, such as the center of the universe, its current expansion, and hypothetical collapse.

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html## Frontiers and Controversies in Astrophysics## ASTR 160 - Lecture 17 - Hubble's Law and the Big Bang (cont.)## Chapter 1. Review of Magnitudes [00:00:00]
Okay, magnitudes. There’s a couple of these magnitude equations. I’m just going to write them down. The first of them looks like this. And this equation is used–okay. So, this equation is used to relate magnitudes of two different objects to each other. So, we’ve got two different objects. And it can be used for either kind of magnitude–either absolute or apparent magnitude, just so long as you don’t mix them. So, it’s two different objects, but only one of the magnitudes. One kind of magnitude. And depending on which kind of magnitude you use, this brightness ratio–it’s either the ratio of how bright it looks or the ratio of how bright it is–whatever’s appropriate. Now, on the help sheet on the web, I have this equation in a somewhat different form, and it’s important to realize that it’s the exact same equation. Watch this. Let’s see. Let’s multiply both halves by - Okay. The other equation looks like this. 5 log ( And that brings me to the comment I want to make about problem 2 And having been asked to do this, the logical thing that you might try to do is say, all right, I’m going to use one or the other of these equations–I’m not sure, in advance, which, to compute this one. Then I’m going to compute this one. And I’m going to subtract the two, and that’s going to give me the answer. That approach will fail. Okay? That doesn’t work in this particular case, because you don’t actually have enough information to compute either one of these things. You do have enough information to compute the difference. And let me just give you a very brief hint on how you might go about doing that. Let’s see. Let me take a new piece of paper here. Write down 1 ‒ 2. So, then, you get Okay, now. Here’s the trick. Turns out for reasons that you had better tell me–and TFs [teaching fellows], take note that we really want them to say why this is true, now that I’ve told them it is. This is zero. The two apparent magnitudes are the same. And so, that means that this side of the equation is what you want. It’s the difference between these two magnitudes. And then, over here, you have to use one of these log rules: log ( And if you use that, it turns out that you have information elsewhere in the problem, which will tell you what you need to know about the distances. And so, in this way, you can solve for the difference without actually being able to determine either one of these two things. So, we’ll just leave it at that for the moment. If you have problems there’s the usual forum, there’s the usual office hours, but ponder this. That’s basically how the problem has to go. Okay? Problems with magnitudes? Okay. If you do have some, let us know, because this is going to be critical for solving, basically, problems for the whole rest of the class. All right. Yes go ahead.
## Chapter 2. Implications of Hubble’s Discoveries on the Aging Universe [00:07:38]Let me remind you why we’re putting ourselves through this pain. Okay? Recall why we started doing this in the first place. The goal was to figure out how to measure the Hubble Diagram. The Hubble Diagram is this diagram of velocity, which you can measure by redshift versus distance. And the whole reason we embarked on this adventure in magnitudes was because that’s a critical component in how you determine the distance. But if you’ve got a bunch of galaxies and you measure these two quantities for each one of them, what you discover–what Hubble discovered–what Edwin Hubble discovered many years ago, is that they line up. You get this perfect; well, not quite perfect, but close to it–this beautiful straight line, if you measure a bunch of these things. And the way you represent that straight line is with this equation, where What I now want to do is talk about the implications of this observational fact that galaxies line up on this line. It turns out, this is one of the most profound plots in all of astrophysics, and possibly all of science. Because what this implies is, first of all, that the Universe is expanding, and hence, it’s the basis for the whole Big Bang Theory of cosmology. And by performing relatively simple calculations using this quantity
Okay, so, here’s what I want to do. I want to start understanding how this plot and this little equation gives you all these wonderful things. I’m going to go on for a little while, then we’ll pause, and we’ll do one of these Q & A sessions, because this is sort of the heart of the Big Bang Theory. And so, we’ll do one of these things that we did when we were talking about relativity, where you talk to each other and come up with questions. So, if you’ve got questions along the way, by all means, ask them, but we will have a specific moment a little ways down the line where we actually pause and do this on purpose. So, everybody, keep thinking as we go along, what are your questions? What don’t you understand or what questions could you ask to understand more than what I’ve just told you? Okay. Here we go. Imagine a one-dimensional Universe, just because it’s easy for me to write down. And here’s our one-dimensional Universe. It’s all strung out on a line. Here’s the line. And it’s got a bunch of galaxies on it. Let’s label these galaxies, Now, next thing we’re going to do: the Universe is going to double in size. So, we’re just going to stretch the thing. The whole thing is going to get stretched. So, here’s our Universe. And now Now, we’re going to ask, if you sit–if you live in galaxy So, galaxy And so on down the line. I could repeat this simple exercise. Now, this is true regardless of which point you sit on. Let’s imagine that we sit on point–that the observer is on point The motion–okay, at the start it’s–let’s take a quick look at how this is set up here. Yes, Let’s look, for example, at So, that’s the key point–that if you take a coordinate system and you expand it, you naturally get this relationship between distance and velocity. Or to turn it around, if you observe this relationship between distance and velocity, then what you’re looking at is a system in which all the coordinates are–in which you’ve simply stretched the coordinate system. Okay. Now, this gives rise to–this analogy, with these stretching one-dimensional lines gives rise to two questions people have, which I like to call un-questions, because they’re actually questions that arise because of the analogy, not because of the way the Universe works. One question is, Q1: “Where is the center?” You know, here’s your line. It’s expanding. But somewhere in the middle here, around So, let me give you a slightly better one. We’ll stick with the one-dimensional Universe, but now we’ll do it this way. Here’s a one-dimensional Universe. You have to stay on the line. So here’s But, notice that this system is unbounded. There’s no edge. There’s no edge. There’s no place where you can say, this is the end of the Universe, because if you traveled around it you’d just come back to where you were, and therefore, there’s also no center. And where does it expand into? It expands into a dimension that, if you’re a one-dimensional creature, you can’t experience because the whole thing is being pushed out. But if you’re forced to live on this circle you can’t even–you have no comprehension of what it expands into. It expands into a higher dimension. But all of this stuff about, you know, velocity and distance, remains basically the same. Here’s a two-dimensional analogy. Let’s see, this is–so, I made a little diagram of the Old Campus [an area of the Yale campus]. Here’s Linsley‒Chittenden [a classroom building] where we’re sitting right now, yes? Here’s the statue of Abraham Pierson. This is the gate between Durfee and Wright [two undergraduate dorm buildings]. Here’s Phelps Gate [a classroom building and the entrance to Old Campus]. Here’s Vanderbilt [a dorm building]. This is Harkness Tower and here’s Starbucks. Okay, that’s all that’s important, right? So, you following me with that? And what I did was, I took this little picture and I took it to the Xerox machine and I blew it up by 20%. So, here’s the exact same diagram blown up by 20%. So, now, supposing we’re sitting in Linsley-Chittenden, which we happen to be doing, and the Universe expanded by 20%–or, our little corner of the Universe expanded by 20%, here’s what would happen. Now, notice what’s happening. Every object in the Universe is moving away from us. See? Here’s where Harkness was, and now it’s moved a little further, in a straight line away from us. Here is Pierson, and he’s moved a little further, straight away from us. And here’s Phelps, and it’s moved a little further, straight away from us. And let me erase those lines, because what I want to demonstrate is that if you’re anywhere else in this Universe, the exact same thing happens. Here we’re now sitting on that statue, and Linsley-Chittenden is moving away from us. Harkness is moving away from us. Phelps is moving away from. Starbucks is moving away from us, and so forth. Similarly, if you’re sitting in Starbucks, waiting for students to come by or something, the exact same thing happens. And now, because the distances are greater, you can see the effect that the velocity is greater at greater distances. If I’m looking down at Vanderbilt, it moves straight away from me, but only a little bit. If I’m looking all the way across Old Campus, this gate moves a lot away from me. And so, once again, you have a situation in which the further away some–everything is moving straight away from you, but the further away it is, the faster it’s moving away, right? And that’s just a consequence of the fact that you have taken this geometry and expanded it. And so, wherever you sit in an expanding geometry, every object you see will be moving directly away from you. And the further away it is, the faster it will be moving, which is Hubble’s law. Oh, and one other thing about this nice analogy, here. Let us imagine for a second that this tiny piece of a tiny Universe is actually not a flat plane, but is sitting on a curved surface, which is curved all the way round into a big ball. That’s actually not so hard to imagine because it’s true. This sits on the surface of the Earth. And so, what is happening when this thing blows up by 20% is, basically, somebody has taken a valve to the Earth and has blown the Earth up by a factor of 20%. And that would have this effect. And it would have the exact same effect everywhere else on the surface of the Earth. And the Earth, you know–where is the center of the surface of the Earth? You can answer the question: “Where is the center of the Earth?” But you can’t answer the question of where is the center of the surface of the Earth. Because wherever you sit, whether you’re sitting at Starbucks or in Phelps Gate or, you know, in Los Angeles somewhere or wherever, if they blow the Earth up by 20% you’re going to see this exact same effect. Everything will be moving away from you. The further away something is the faster it will be moving. So that’s the one–yes, go ahead.
## Chapter 3. Conceptualizing a Three-Dimensional Universe [00:26:36]Here’s the thing I want to do. So, we’ve had the one-dimensional case, the circle. We’ve had the two-dimensional case, the expanding sphere. Of course, what we want is the three-dimensional space. Okay, here we are in three dimensions. Someone is expanding the Universe, so everywhere we look, everything is going away from us, and the further away it is, the faster it’s going. What’s it expanding into? Well, that, we have a little more trouble visualizing, right? Because in one dimension, you can visualize this circle expanding onto the plane. In two dimensions, you can imagine this spherical surface expanding. In three dimensions, we can’t imagine what it’s expanding into. That’s beyond us. And so, having had this failure of the imagination, what do you do? You resort to mathematics. That’s what we always do. And so, imagine that every object has a position, which is denoted by three coordinates, three spatial coordinates And there are two ways that things can change their position. One is, they can move; they can change their But the other is just the effect of the change. And in particular, in the case of the current Universe, the increase in the scale factor. And these two kinds of velocity are conceptually different from each other. Because you don’t have to do anything to change your position in this way. You just sit there. You don’t expend any energy. You don’t have any requirement to expend energy or to exert a force or to do any of these things that we ordinarily do to change our position. You just sit there and the Universe expands you, or expands your position. And that’s why, going back to your question, that’s why it’s possible for this kind of velocity to turn out to be greater than the speed of light, if it’s far enough away, whereas, it’s not possible here. What the effect of having this kind of velocity be faster than the speed of light does, is it makes the object impossible to see, because photons coming off them would be redshifted into oblivion. And so, you can’t actually see them. And this imposes a kind of cosmic event horizon, similar in kind to the event horizons around black holes. You can’t see events on the other side of these horizons. All right, let me go backwards in time. Back in time to when this scale factor And yet, the whole geometry of the Earth, of the Old Campus, whatever, is already somehow encoded in that point. Because, you know, you’re taking 0 times So, it is wrong to think of the Big Bang as starting at a point and expanding into space. That’s kind of the impression that word the Big Bang gives you. And you think, naturally enough, of an explosion, where something at a point explodes into empty space. But that’s not right. All the space, all the empty space is contained in that point. It’s all in there. It’s just, it’s all multiplied by 0, so it all comes out to be on top of each other in the same place. How are we doing? Wonderful. Okay, that’s the essence of the Big Bang–the whole idea of the Big Bang–that what is happening is that the whole coordinate system of the Universe is multiplied by this constant. And that constant changes in time. It gets bigger. And that fact is inferred from the observations of these galaxies. That you observe galaxies, and that they’re moving away from us. But more than that–that there’s a linear relationship between how far away from us they are and how fast they’re moving. That implies this kind of coordinate expansion. And if you run it in reverse, it implies an origin to the Universe at some specific time in the past. ## Chapter 4. Q&A: The Big Bang, the Expansion, and the Big Crunch [00:34:22]All right. Let’s have questions. I’ll tell you what. Talk to each other. Come up with good questions, and remember how we do this. When you come up with a question–a question can be either to explain some of this or to expand upon it. When you’re ready with a question, put your hand up. I’ll answer a few of them while other people are getting their questions together. And then we’ll answer as many of them as we have time for in the remainder of the class. So, talk to each other. Talk amongst yourselves. Come up with a question, any question. All questions are good, and see what you can do. I’ll come around and try and answer some of these. Yeah ‒ so, talk to each other by all means. Yes?
Okay, let’s have a few of these. Yes, go ahead.
And there’s basically two ways to think about this. One is by analogy. You think of the one-dimensional creature and what it thinks about the second dimension, or the two-dimensional creature–what it thinks about the third dimension. And then, you just, sort of, take this leap of faith into us moving into the fourth. Or, you have to write things down mathematically. Those are the only two options you have. Actually picturing this is not going to happen. Yes?
No. Here’s why. We are not–we–that is to say, you, me, our bodies, whatever–are not participating in the expansion of the Universe at the moment. Why? Because the molecules in our body are being held together by other forces–chemical forces. We are being–the Earth is held together by gravitational forces and those forces have stopped the expansion of the Universe locally, but not globally. And that’s why you have to go out and measure this stuff with galaxies. Otherwise, you’d be able to, you know, measure it with this. Now, of course, then, there’s the problem, the ruler also expands, all this kind of stuff. But, in fact, what happens is that locally–local objects, by which I mean, our own galaxy and anything smaller, are held together by other forces. Picture a balloon expanding, so that’s the two-dimensional case, this balloon expands. And put a bunch of leather patches on the balloon. So, those leather patches don’t expand, but the distance between–because they’re held together by something else–but the distance between them does. Yes?
Why? Why is it increasing by the particular amount it is? You have to think of that as being a parameter of the Universe. It’s one of the things about the–sort of like the speed of light. Why is that the quantity it is? Or why is the gravitational constant the value it is? It’s one of the parameters that governs our Universe. Why those particular parameters? That’s not quite a science question, and we don’t know. You can turn it around. You can make the following interesting point. If it wasn’t expanding at approximately the rate it is, we wouldn’t be here to notice it. Because if it was expanding a whole lot slower, then it would never have gotten big enough for stars to condense out. If it was expanding a whole lot faster, all the atoms would be spread out so far that, again, stars could never form. So, you can make this interesting argument that while you don’t know why it has this particular value, it’s very important that it does, because otherwise we wouldn’t be here to observe it. This is a form of argumentation called the anthropic principle. It is highly debated among scientists whether this is a scientific argument or not. But it’s amusing either way. So–yes?
This, by the way, is why the Catholics like the Big Bang Theory so much. In fact, the mathematics, the relativistic mathematics that describe the Big Bang and the expansion of the Universe, were worked out by a man named Lemaitre, who was a Catholic priest, who was a Jesuit. They love this because it gives you a creation moment. It kind of gives you a scientifically verified creation moment. John Paul was very enthusiastic about astrophysics. He used to throw big conferences in the Vatican, give after-dinner speeches. There’s a thing called the Vatican Astrophysical Observatory. They have a bunch of Jesuits. They run a telescope in Arizona. They do research into this. And so, you know, if you want to see science and religion converge, you want to run away from biology as fast you can and talk to us about cosmology. And what I don’t understand is why the, sort of, fundamentalist type worry so much about biology, where all the science is dead-set against them. Whereas, in the case of astrophysics–now, of course, it is possible, if you take an atheistic point of view, to come up with all kinds of clever ways to avoid this creation event. But again, it stops being science at a certain point. It becomes another odd kind of theology, and we’ll talk about that, perhaps, a little bit. But if you want a place where science turned out to be the congruent to, at least, certain kinds of non-fundamentalist religious beliefs, you’re way better off in astrophysics than you are in biology. Yeah, go ahead.
And so, what you expect is the Universe is slowing down. Because you know you don’t know how much it’s slowing down, whether it will stop the expansion or not, but you expect it to be slowing down. The punch line of this whole part of the course, in order to anticipate where we’re going, is that, in fact, we can measure the change in the expansion rate, and it’s speeding up, not slowing down. This is very disturbing, because it means there’s some kind of pervasive cosmic anti-gravity, which we call dark energy, because we don’t know what it is. And that’s basically the punch line of this whole section of the course. But you could imagine that there is a change in the expansion rate, either positive or negative, and that, therefore, various different outcomes are possible. Yeah?
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