Frontiers and Controversies in Astrophysics

ASTR 160 - Lecture 18 - Hubble's Law and the Big Bang (cont.)

Chapter 1. The Expanding Universe ‒ Big Bang and Steady State Theories [00:00:00]

Professor Charles Bailyn: We’ve been talking about the expansion of the Universe and in answer–I want to follow up on something I said kind of in answer to a question last time, which is that it turns out that the expansion of the Universe alone doesn’t require you to have a Big Bang. There are other explanations, or in particular, one other class of explanation, which is compatible with the observation of the expanding Universe, but doesn’t exactly lead to a Big Bang, in the sense that we think of it today. And so, I want to talk about these alternatives.

This is Frontiers and Controversies as of 1950. In 1920, you’ll recall, they were worried about whether the spiral nebulae were island galaxies of their own, or not. That was settled by Hubble’s observation. In the 1950s, it had become clear that it really is true that the Universe is expanding. But there were two categories of explanation that were being put forward to explain that, and deciding which is which was the current big topic of the day. So, the Universe expands. And what do we make of this fact?

Well, one option is the Big Bang, what we now call the Big Bang, which, as I’ve mentioned, implies that in the past, everything was closer together. It was denser, and that creates other changes. If things get denser, they also get hotter, you may remember from chemistry. So, this implies in the past things were different from how they are now. They were denser, hotter, and that in the future, it’ll go the other way. Things will become sparser and cooler. And there may be other changes associated with this. There may be different kinds of galaxies in the past from now and different kinds of galaxies now, as compared to the future, and so forth. But the Universe is a place whose bulk properties can change in time. So, things change in time.

And the implication is that you can extrapolate this back to an initial singularity–that there was a moment, at some point in the past, where all currently existing space was piled up in a single point. The way I talked about this last time was that the scale factor was equal to zero, and so, there is this implication of an initial singularity. Now, the initial singularity is not the kind of thing that can be verified scientifically, because all the physical laws break down the same way they do in the singularity inside a black hole, or in an event horizon, or something like that. But nevertheless, that’s the implication. That’s the extrapolation of this set of ideas.

But, at least, at the time, that was not the only set of ideas that could explain this expansion. There was an alternative, which was described as the “steady state.” I should say that both of these names were given to these ideas by people who supported the “steady state.” Big Bang was–that phrase was used as an insult by the “steady state” people to make fun of the ideas of the Big Bang, which we now actually know to be correct. And that’s why that particular phrase is, in a way, so misleading–that they were deliberately trying to obscure and insult some of the ideas in the Big Bang. From time to time, astronomical organizations and popular magazines run contests to rename the great theory and come up with some better name that won’t give you in your mind the impression of an explosion coming out from some central point. And people send in suggestions and it never gets anywhere. We’re stuck with the Big Bang. But it is good to remember that it was named by its opponents, who supported the “steady state.”

So, how would this work? The central idea here is that the Universe expands, but then, new matter and energy is created to fill in the voids, as the Universe expands. So, sure, all galaxies are moving away from us. But new galaxies are being created where the old galaxies used to be. And the consequence of that is that the past and the future both have similar bulk characteristics. The density, the overall density, is the same. And other characteristics are the same, because as things move further away, you’ve simply replaced them with other things like them.

And so, now, you have to kind of invent some way of creating matter and energy. But that’s not nearly as bad as creating a whole Universe, which is what you have to do in the case of the Big Bang. So, in this case, the past and the future are the same.

It also implies that the Universe is eternal in contrast to the Big Bang, where there is this initial moment, which one then instantly asks, well, how did that happen and what happened before it? So, we get around that. This is an eternal process. And it’s also infinite. And so, we don’t have to start asking these awkward questions about what’s outside the Universe. And so, people found this, in some ways, much more satisfying.

And so, there were these two quite different explanations of the observed fact that the Universe was expanding. Now, the historians of science, who have studied this controversy, some of whom make the claim that which side you ended up on in the 1950s, before there was a lot of evidence either way, this was kind of a philosophical rather than a scientific question at the time–that which side you ended up, here, depended largely on your religious beliefs or lack thereof. And they document this–a lot of the work on the Big Bang and a lot of the supporters of the Big Bang were, as I mentioned last time, Catholics. In one important case, this guy Lemaitre, who worked out all the equations, was in fact a Catholic priest, also a physicist in his spare time.

And this is a nice kind of idea, because from that point of view–because it gives you this initial moment of creation, which is–if you’re a religious person, it’s relatively easy to come up with a reason why that might have happened. On the other hand, a lot of the scientists, particularly in Britain, where the “steady state” was largely developed, were of a kind of atheistic or agnostic turn of mind, and they didn’t like this initial singularity. And they liked the idea of an eternal infinite Universe, as actually Einstein did, too, because you didn’t have to have a creation event, and you didn’t have to invoke the kinds of ideas that come along with that. And there’s some dispute over how much this really mattered to people.

But what was clear is in the 1950s, this was not a question for which there was scientific evidence either way. And so, what the scientific evidence said was that the Universe expanded. And then, you could follow either of these arrows. Interestingly, however, these two hypotheses make quite different predictions, which in the end, turned out to be testable. And, in particular, what it predicted, if the Big Bang makes this very strong prediction that the past is fundamentally different-looking from the present. Now, over the course of a human lifetime, this isn’t going to make a lot of difference. The galaxies don’t get that much further away from us in twenty or thirty years.

But one can actually look into the past by looking at things that are far away. This is because the speed of light is finite. When we look at the Sun, we don’t see the Sun as it is right now. We see the Sun as it was eight minutes ago, when the light started to travel toward us, because the Sun is eight light minutes away. If we look at Alpha Centauri, the nearest star other than the Sun, if Alpha Centauri blew up or disappeared or something right now, we wouldn’t know about it for four years because it’s four light years away. Similarly, if we look at a galaxy that’s 10 megaparsecs away, a parsec is three light years, so a megaparsec is three million light years. Some galaxy that’s 10 megaparsecs away, that’s 30 million light years away. So, we’re seeing it not as it is, but as it was 30 million years ago.

Chapter 2. Quasars and the Rejection of the Steady State Theory [00:09:53]

Now, it turns out that in the course of cosmic time, a few million years here or there doesn’t make any difference to anybody. But once you get into billions, now you’re talking cosmic time. And so, if you look at things that are billions of light years away, that turns out to be a substantial fraction of the age of the Universe predicted by the Big Bang. We’ll get back a little later in this lecture to how you determine what the age of the Universe is supposed to be. But if you go back a substantial fraction of the age of the Universe, if you look back over distances that great, then you predict that you ought to see things that look different from the galaxies you see today, because the Universe at that time was denser and hotter, and the galaxies were younger on average, and you ought to be able to see some kind of differences.

So, the past is different from the present, and this is observable through this concept of the “lookback time.” Just the fact that the further away something is, the longer it’s taken the light to travel to you. And so, you’re actually seeing things not as they are right now, but as they are in the past. And so, you can imagine a great research project where you sort of look at nearby galaxies to see how things are in the current day. Then, you look at really, really distant ones, some substantial fraction of the age of the Universe ago, and you ask yourself the question, “Are those galaxies the same as the galaxies that exist today, or are they, in some sense, different?” So that’s a testable prediction.

There’s also a testable prediction on this side, which is that you ought to actually be able to find these places where the new matter and energy is being created, because that’s a constant ongoing process. It has to be, because it’s got to fill in the empty spaces left behind by the expanding galaxies. So, the prediction over here is that there exist places of matter and energy creation.

So, this is potentially testable. And in the 1960s, evidence was actually found that kind of decided this question, which is why we now no longer believe in the “steady state.” And two things happened; actually, there were a number of things that happened. But one of them was the discovery of what were called “quasars.” These are now known to be accreting supermassive black holes, but they didn’t know that, then. So, this is in the 1960s. That’s what we think they are now, but they didn’t know it then. All they knew is that they had discovered a great source of huge amounts of energy.

And you’d think that this would be a very good thing–some kind of unknown energy source would be just the thing for the “steady state” people, because that’s what they needed. They needed some kind of source of mass energy creation, and indeed, that was claimed for a little while.

The main characteristic of these things is that they have very high redshifts, which implies very large distances, because velocity is proportional to distance. That’s the Hubble Law. Everybody agreed that the Hubble Law was right. So, these things were known to be large distances away. And so, if you’re a “steady state” person, you say, well, that’s great. If they’re so far away, they must be incredibly bright, otherwise we wouldn’t be able to see them, and there’s our energy source that we require.

If you’re a Big Bang person, what you say is, now we’ve finally found a bunch of things that are really far away and we can ask the question, “Are things that are that far away the same as in the local Universe, or are they different?” And what it turned out–one of the very first things that was discovered is that there were many more quasars in the past than there are now. And so, whatever they are, supermassive black holes or whatever hypothesis you have for these things, it turns out, they’re dying out. There were more of them in the past then there are now. They were brighter.

And so, here is an example of a significant change in the composition of the Universe, which is the thing that’s predicted by the Big Bang. And so, although initially, it looked like quasars might be helpful to the “steady state,” as soon as they started to get enough of them so you could do these kinds of statistical tests, it became the first really strong evidence that the Universe was changing in time. This was not the only thing that was discovered in the 1960s. Another thing is the so-called cosmic microwave background, and we’ll talk much more about that later on.

Right now, I just want to say that what this is, is it’s radiation that was created–that was generated when the Universe was much denser than it is now, much hotter than it is now. In fact, it comes from radiation emitted by ionized hydrogen. So, this is when the whole Universe was at 10,000 degrees or so Kelvin–obviously much hotter than the Universe is right now, and much smoother. All this hydrogen, rather than being locked up in individual stars or in–and stars concentrated in individual galaxies, all this hydrogen was very smoothly distributed around space. And so, this came from a time–this radiation, which you can easily detect with radio telescopes and the like, comes from a time when then Universe was much denser, hotter, and smoother than it is today–again, in accordance with the prediction of the Big Bang.

So, this provided very strong support for the Big Bang. And the discoverers of this have got–there have been Nobel prizes given out on a regular basis for discoveries related to the cosmic microwave background. And as I say, we’ll come back. I’ll talk about that at some length later on.

Another thing that was discovered is that the Universe is pretty clearly, mostly, ¾ hydrogen and ¼ helium. Just about every cosmic object you see, not the Earth, but the Solar System as a whole, has those proportions. And all stars have these proportions and all galaxies consist of stars that have these proportions. And it turns out, this could be explained by the Big Bang. When it was denser, hotter, and smoother in the early Universe, it was hot enough for hydrogen fusion–hydrogen fusion into helium.

But it didn’t last very long because the Universe was expanding and cooling, and you need to have very high temperatures in order to have hydrogen fusion. That’s why it can only happen in the center of the Sun, or in an atomic bomb explosion, or something like that. And so, the Universe, for about three minutes, was hot enough for hydrogen to fuse. And if you ask yourself, what fraction of the hydrogen fused during the time the Universe was hot enough for that to occur, the answer turns out to be, about a quarter of it–in exact agreement with the currently observed fractions of hydrogen and helium in the Universe.

So, this is in first three minutes. There’s a famous popular book from the 1980s called The First Three Minutes, which discusses this, by Steven Weinberg. In the first three minutes, one quarter of hydrogen fuses into helium, and afterwards, no more, because it’s too cool for these reactions to occur. So, that was another piece of evidence that the Big Bang was the right explanation and the “steady state” was not. Yes, question?

Student: Is it actually three minutes or is that just [Inaudible]

Professor Charles Bailyn: Sorry.

Student: It is actually three minutes?

Professor Charles Bailyn: So, if you start from time zero, when everything is put together, after three minutes the Universe has cooled sufficiently so that you can’t have hydrogen fusion, generally. Then, no more nuclear reactions occur until much, much later, when stars formed. And, we know how fast the Universe is expanding because we can measure the Hubble Constant now. And so, you know how long it takes for the Universe to expand to the point where the temperature drops enough so that there’s no general hydrogen fusion anymore. And then, you do this little calculation to how much fuses during that time. Other questions?

Okay. More recently, with the Hubble Space Telescope and other big ground-based telescopes, we’ve discovered that you can now see galaxies that are very far away, that are as they were a substantial fraction of the age of the Universe ago. And, by now, it’s clear that galaxies evolve and that the statistical demographics of galaxies–how many there are, how massive they are, how big they are, all those kinds of things–that those statistics change dramatically over the course of the Universe–over the course of the amount of lookback time that we can currently observe. So, galaxies evolve very significantly.

So, everything points toward the Big Bang and away from the “steady state,” much to the surprise of a lot of the scientists who felt, possibly correctly, given what was known in the 1950s, that the “steady state” was a much more elegant solution. And so, let’s see. The fable here is the demise of the “steady state.” And the moral, well, there are various morals to this, but let me be provocative and say that, sometimes science is anti-atheistic, not anti-religious.

Because it could have come out the other way, right? It could have turned out that this “steady state” was right. And then, you wouldn’t have had this moment of creation and you wouldn’t–and Pope John Paul II wouldn’t have been so enthusiastic about astrophysics. In just the same way, it might have turned out that human beings are fundamentally different from all other species of animals. And if that had been true, the geneticists would have discovered that instead of what they did discover. And so, there’s a feeling, particularly in current political debates, that scientists are somehow intrinsically anti-religious, and I think that isn’t true. It isn’t one way or another. It’s just a way of finding stuff out. And sometimes, as in this particular case, you might find out that the atheists, or whatever the atheists had proposed, is wrong. And that’s actually what happened in this particular case.

So, the demise of the “steady state” is one of the big things that happened in the latter half of the twentieth century. And so, now, the scientific evidence is very, very strongly in favor of the Big Bang. I should define what I mean, I think, by the Big Bang Theory, because, as I said, this is a phrase coined by its enemies. And it’s one of these phrases like “black hole,” which actually doesn’t have a technical definition, and you get into trouble by people using it in different ways.

And so, if what you mean by this is that, in the past, the Universe was denser and hotter, and smoother–because it’s constantly forming new stars and galaxies. Everything is getting more lumpy with time–and that you can extrapolate this to an initial singularity, then, I think there’s extremely strong scientific evidence in favor of this. But you have to keep your eye on this word “extrapolate,” because sometimes the Big Bang is used to describe this moment of initial creation, or whatever you want to call it. This initial moment of infinite density, and somehow everything starting to expand. You can’t observe that. That’s one of these things that you can’t see now, or with our current theories at any point in the future. And so, that’s an extrapolation. That’s not an observation. And that’s a fundamentally different thing. So, if you use Big Bang to indicate that particular moment, then you can say that the scientific evidence for that actual moment isn’t strong, and indeed, couldn’t even exist within our current theories, because all of physics breaks down, there.

And so, what I mean when I say that there’s a lot of scientific support for the Big Bang is, there is support for this idea that the Universe is changing in time. It was denser, hotter, and smoother in the past, and that if you extrapolate this back, there was an initial singularity some number of years ago. And you can actually figure out how many years ago that was, which is what I’m about to do next.

And so, this is now the kind of currently accepted theory. I should say, there are a few remnant holdouts from the “steady state” type, who annoy the rest of us by not giving in. And there’s actually been some interesting controversy over the years about, you know, how much telescope time do you give to people whose proposal is to look for the places where mass is created in the “steady state” theory, if nobody else believes that theory anymore. And, by now, the answer is none, but it took quite a while to get to that point. Okay.

Student: Do people ever lie about what they’re going to observe?

Professor Charles Bailyn: Do people lie about what they’re going to observe? Excellent–the graduate students are amused. No. What happens is this. You have to present–so, you have to write these proposals, because the telescopes are over-subscribed. And you have to come up with a plausible thing that you’re going to do. Now, it then varies how this is actually done in operation. In the space telescope, for example, what you have to do is, then, you fill out what’s called a Phase II form, which tells exactly where the space telescope is supposed to point and for how long. And you submit that, and they upload it and they do it. If you deviated drastically from the target list you gave them when they approved your proposal, that’ll get flagged and they won’t do it. On the other hand, on ground-based telescopes, you kind of go to the telescope, and still, in many cases, you operate it yourself. And there’s not a lot of control over where you point the thing. The control then happens later.

One of the key components of a proposal for telescope time is what you did with the data you got from the last amount of telescope time they gave you. And, you know, you have to list all the publications, or if you haven’t actually gotten to publishing anything, which is usually the case with me recently, you have to show little graphs or, you know, describe the data and what you’re going to do with it and so forth.

And one of the things–I sit on these committees that make these kinds of decisions. And one of the things you look for is if they haven’t done interesting work and, kind of, done what they claimed they would do the last time round, their proposal goes down to the bottom. You’re always looking for ways to trash other people’s proposals, because you’ve got seven times more–in the case of the space telescope, you’ve got seven times more proposals than you can grant, of which only a small handful are not worth doing. And so, any opportunity you have to say, you know, these guys are bozos–you definitely take that opportunity, because otherwise you have way too many good proposals left over.

So, there’s a kind of internal control that isn’t explicit on this sort of thing. And after a while, you know, if people keep getting up in public and saying, you know, quasars are sources of mass energy creation and therefore support the “steady state”–even if they’re a great big quasar expert, you start to get a little bit queasy about giving them large amounts of telescope time that might be more profitably used by someone else.

This, then, gets interpreted by the remnant “steady state” supporters, or whoever the minority idea might be, of a hugely oppressive scientific bureaucracy, you know, not allowing the maverick, wonderful thinker to do their own thing. And that, sometimes, is true, but not often. Most of the time, it’s the sane people not allowing the insane people to use the telescopes, and that’s actually a much more common thing. And so, while it can be good propaganda to say, yeah, these oppressive, elitist, bureaucratic people–mafia who run the scientific world are not allowing my great idea to get any opportunities to prove itself. Most of the time, it turns out, the establishment is right. And so, there is this interesting question about allocation of resources. It’s not just telescope time, more importantly, even, money.

There’s a big debate right now, for example, in the theoretical physics community about string theory. String theory is the hot theory of everything. And everybody is supposed to be a string theorist, in theoretical physics, at the moment, except for a few people who point out that it hasn’t actually been all that successful in explaining anything. In fact, it hasn’t explained anything, ever. And therefore, might one not want to consider alternative theories?

And then, the string theorists say, but, you know, this is a really good idea. We’ve got to continue to look at it. One day we’ll get it right and we’ll figure out everything. And so, there has now, recently, been some popular books that claim that string theorists are oppressing everybody else by not letting other kinds of good ideas be funded, and by not letting smart young people who are working on other things–by not hiring them into departments, and so forth.

It’s an interesting argument, and ongoing, in the string theory community. It’s, sort of–as I said–they’ve been writing popular books on both sides of this, so, it’s kind of busted out into–you’re going to read about this, now, in the New York Review of Books, and places like that. And so, what you do about minority ideas–ideas that are not supported by the current paradigm of a given scientific subject is a very tricky one, and a very interesting one, and one that needs to be reevaluated from time to time.

And, in fact, cosmology may be approaching such a moment, where a genuinely new idea is going to have to be required. We’re probably not quite there yet, but we’ll talk about alternative cosmological theories toward the end of the course. Sorry, I didn’t mean to go off on this, but it’s on my mind, because, as I said, I serve on these committees, so you have to think about these things.

Actually, the space telescope people did a very interesting thing. At one point they decided–I don’t think they ever actually followed through on this–that 5%, or some small amount of the time–different scientific resources sometimes have this‒that, like, 5% of the time goes for risky science. Science that’s really weird, and probably won’t work–but if it works, it’s incredibly important because committees tend to not to want to do that. They tend to want to do the things that they know are going to produce some good result. And so, sometimes, the people who organize these things force the committees to have a little category of special, weird projects.

And then, of course, what happens? Five years later, they analyze, you know, where did all the good science come from? And if it didn’t come from the weird projects, which is almost certainly the case–although not 100% certainly the case, but generally the case–then they say, look we’ve wasted 5% of our money, telescope time, whatever it is. We’re going to close down this program. And then, you kind of have to start over again. So, how weird is weird? Difficult to say.

Chapter 3. Calculating the Duration of the Big Bang [00:32:21]

All right. The expansion of the Universe. We now believe, for the reasons I outlined, that this indicates a kind of Big Bang idea, which means we can extrapolate back to the moment that it all started, and we can calculate how long it took.

Let me do a simpler calculation, but exactly analogous. Supposing you’re in a car. You’re in a car and you’re driving at a speed of 50 miles per hour. And you are 100 miles away from your starting point. How long have you been driving? So, this, they taught you how to solve in seventh grade. If you are 100 miles away, you take time is equal to distance over velocity [t=D/V]. So 100 miles. You’re going 50 miles an hour. That means you must have been going for two hours. Not such a hard problem. There’s a hidden assumption though–sorry?

Student: Speed is constant?

Professor Charles Bailyn: Speed is constant, yeah–provided that the speed is constant. Okay. But let’s make that assumption. Let’s assume that the Universe is expanding at the same rate all the time. How long has it been going? Well, take some galaxy, any galaxy. It turns out, it doesn’t matter which galaxy. And say–so, galaxy A is at distance D. It’s moving at velocity V, away from us. You know, that’s the whole idea. So, how long has it been since it was right on top of us? Well, so, it’s been going for a time equal to D/V. By exact analogy with the distance, velocity, time, questions that they ask you about–you know, cars driving to Cleveland, and stuff. So, it’s been going for time, T= D/V.

But now, in the case of galaxies, there’s this interesting relationship–that V = H/D. So, D/V = 1/H. Okay? And that can be measured. That’s Hubble’s Constant. And so, the age of the Universe is equal to 1 over the Hubble Constant, provided that the expansion has been at a constant rate.

Let me remind you, H has been measured. It’s 70 kilometers per second per megaparsec. That’s kind of an interesting unit. You’ve got kilometers per megaparsec. Both of those are measures of distance. This is basically velocity over distance, but velocity contains a distance within it. So, if you were to cancel those two distance terms in some way, you’d have that the units of the Hubble Constant are 1 over seconds. So, the units of 1 over the Hubble Constant are seconds. It’s a measure of time. It’s reciprocal time. 1/H is a measure of time.

What is that time? It’s the age of the Universe. What do I mean by the age of the Universe? That’s how long it has been since all galaxies and all objects in the Universe, all points in the Universe, were piled on top of one another. Yes?

Student: [Inaudible]

Professor Charles Bailyn: Let’s see, did I write it wrong? V equals ‒ yeah. I wrote it wrong. V = HD. Thank you. And so, now, what am I going to do? I’m going to divide both sides by H. I’m going to divide both sides by V, and then it comes out right. Sorry, thank you very much. Stop me when I do that. Right, because this is the Hubble Law.

Okay. So, how old is the Universe? Well it’s 1/70 kilometers per second per megaparsec. That actually is not a very useful unit of time. So, let’s see if we can do better. H is equal to 70 kilometers per second per megaparsec. Now, we want to cancel the kilometers per megaparsec. So, what we want to multiply this by is the number of megaparsecs in 1 kilometer. That’s a very small number, right? It’s some tiny fraction, because a megaparsec is huge. It’s, you know, three million light-years, or something like that, and a kilometer, not so huge. So, let’s calculate this term. That’s going to be–1 kilometer is 10³ meters. One megaparsec–a mega is 10⁶, so parsec is 3 x 10¹⁶ meters. And so, this is equal to ⅓ times–10^-22–10^-19 equals 3 x 10^-20, right?

But it’s 70 of those. So, H is equal to 7 x 10¹ x 3 x10^-20, which is equal to 20 x 10^-19, or 2 x 10^-18. In units of 1 over seconds. So, 1/H, we know now, in seconds, is equal to 1 / (2 x 10^-18 seconds). That’s (½) x 10¹⁸ or 5 x 10¹⁷ seconds. One year is equal to 3 x 10⁷ seconds. So, the age of the Universe in years is (5 x 10¹⁷) / (3 x 10⁷), which is something like 1.7 x 10¹⁰, or 17 billion years.

So, that’s the answer. The Universe is 17 billion years old.

It’s not quite the answer because of that assumption, right? The assumption was that the Universe is expanding at the same rate all the time, and that’s not necessarily true. And the next thing we’re going to do is ask the question, “What happens if the expansion rate of the Universe changes with time?”

It probably does. It almost certainly does, because the Universe is filled with mass. What does mass do? Mass exerts–well, Newton would have it that mass exerts gravitational force. It slows down the expansion of the Universe. So, what do you expect? You expect the Universe to be slowing down gradually.

And this gives rise to a couple of possibilities, and I think I mentioned this briefly in passing. Now, we’re going to do a more–in more detail. This is the scale factor of the Universe, which started at 0. Here’s time. Here’s now. Here’s whatever the scale factor of the Universe is now, and what has happened. Here’s what it looks like if there’s no change. So, this is constant expansion. And so, it is this amount of time that turns out to be 17 billion years.

But in fact, it seems likely that the Universe is, in fact, slowing down. So, what does that look like? Well, that means that in the future it’s going to do this. That means that in the past, it was doing this, because it’s presumably been slowing down since the very beginning. And that means that the Universe started later than you think, because it started expanding more quickly than it is now, and then it’s been slowing down. So, in fact, you might say that what we’ve really shown is that the Universe has to be less than 17 billion years old.

By the way, this calculation is why it was such bad news when Hubble got the answer wrong. You will recall from the problem set, Hubble thought that the Hubble Constant was 500 kilometers per second per megaparsec. 500 is a bigger number. It means–it’s what? Seven times bigger than we currently believe. That means that Hubble’s estimate of the age of the Universe was a factor of 7 smaller than 17 billion years, which is about 2.5 billion years.

So, the implication of Hubble’s result in the Big Bang context, at least, was that the Universe was 2.5 billion years old. That was very bad news, because the geologists had already figured that the Earth was 4 billion years old. And it’s not good news to have a planet that’s a factor of two older than the Universe. And so, this was part of the initial suspicion of the Big Bang explanation. Part of the reason people tried to develop “steady state” ideas was that Hubble’s calculation of the constant gave you an age of the Universe that was in contradiction to existing geological ages measured on Earth.

By now, we’re up to 17 billion years and that seems to–that’s fine. The Earth is only 4.5 billion years old, so there’s no conflict there anymore. There are some star clusters that are known to be 12 or 13 billion years old, and they caused some trouble for some values of the Hubble Constant until quite recently.

Chapter 4. Calculating the Potential Future of the Universe [00:43:16]

Okay. But then, this begs a question; namely, “What’s going to happen in the future?” Could it be that the expansion of the Universe stops, turns around, and comes back, leading to a “big crunch,” another one of these technical terms? And how much mass would the Universe have to contain for that to be true? So, let me do that calculation now.

This is something we know how to do. This is an escape velocity problem. You know, you throw a pen up in the air. If you throw it slowly, it stops, turns around, and comes back. If you throw it fast, it goes away. Or, to put it a different way, if you throw a pen up in the air at some velocity, whether it comes back or not depends on the gravitational force of the planet you’re standing on, which, in turn, depends on the mass and the density of that planet. So, if you set your Universe into motion, it’s expanding outward. Depending on the density of the Universe, it can stop, turn around, and fall back, or it can keep going.

So here’s the calculation. Here’s us, or any other observer. It works no matter what you do. Then there’s some galaxy, some distance, D, away from us. It’s moving outwards at a rate of some velocity, V, where V is related to D by this equation–second per megaparsec, that’s right.

And then, what’s trying to pull it back? Well, the mass inside of this region is trying to hold it back. It’s only the mass inside of that region because the mass outside of the–we talked about this in the context of planets. This all cancels out.

And so, what is the escape velocity? So, the escape velocity, you may recall, is the square root of what? 2 GM/D. That’s the escape velocity of this object. Let me make sure I haven’t screwed this up. No, that’s right. And so, the question is, “IsV greater than V_escape?”

All right. So, let’s evaluate this. What’s M? M is equal to the density times the volume. Density–we’ll call that–give it its usual symbol of ρ. And the volume is ⁴⁄₃  π D³. That’s the volume of this sphere of material. And you’ll see why I translated into density in a minute. The velocity is equal to H times D. So, the question is: is H times D greater than the square root of 2 G/D times the mass, ρ ⁴⁄₃  πD³? All right, let’s square both sides. I’ve got to get rid of this square root sign. So H²D². IsH²D² greater than 2 G ρ ⁴⁄₃  π? And we’ve got D³ / D, so that’s D².

And here’s the key thing: the Ds cancel. The distance cancels. It doesn’t matter which galaxy you pick. It doesn’t matter how far away that galaxy is. You get the exact same result for every portion of the Universe you examine. So, the question becomes, “Is the density less than 3H² / 8 π G?” I’ve just rearranged the terms. This is the ρ. It’s still on the less-than side. I’ve taken the– 2 x 4 = 8, that’s this 8. There’s a π, that goes down here. This 3 comes over on top, and the H²stays where it is.

So, this quantity is defined as the critical density. If the density of the Universe is less than the critical density, the Universe expands forever. If the density of the Universe is greater than the critical density, then the Universe re-collapses. So, we can calculate this. This is all constants on this right-hand side. We’ve measured H, so we know what that is.

3 x 2 x 10^-18, that’s H. We worked that out, squared, over 8 π 7 x 10^-11.

And now, it’s just arithmetic. And I’ll tell you the answer. You can work it out for yourself. I got 6 x 10^-27, and this is in units of kilograms per meter cubed. That’s a really small density. It doesn’t take much to hold the Universe back.

On the other hand, the Universe is really big. Obviously, the density of this room, of the air in this room is, you know, 30 orders of magnitude bigger. That’s why the Earth isn’t expanding, because we’re so dense that our little region of the Universe has turned around and re-collapsed long since. But if you take the Universe, as a whole, planets are rare. Stars are rare. Galaxies, even, are rare.

And it turns out, as we’ll discuss next time, that the Universe is actually fairly close in bulk. If you take a large enough region that includes all the empty space, it turns out that the average density of the Universe is actually surprisingly close to this critical value at which the Universe will be balanced between expanding forever and re-collapsing. And so, we’ll talk about how you actually measure the average density of the Universe, and thus, determine the fate of the Universe, next time.

[end of transcript]

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