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Professor Charles Bailyn: Welcome back for more cosmology on a rainy day. You'll recall, maybe, what we were trying to do. We're trying to determine the past and future of the Universe — which, it turns out, can be summarized by a single number, namely, the scale factor of the Universe.

It's actually not a single number. It's actually a function. Right? It's a function of time. The scale factor changes with time. So, when I make this grandiose statement that I'm going to talk about the past and future of the Universe, what I really mean is, I want to determine what the scale factor is as a function of time.

And so, that's just a standard kind of a graph. Here's the scale factor, usually given the letter a. Function of t. Here's time. The present is somewhere along this graph. Call that now. And so, here's — and the Universe has some scale factor at the present time. And what we're going to do is we're going to call that — define the scale factor at the present time to be 1. We're just going to — that's called normalizing: when you set — when you multiply something by some convenient number so that it comes out to be 1 at a convenient moment. So, that's just a definition. We define the current scale factor to be 1, to be unity. And so, there's a point here at — right now, the scale factor is 1.

We also know what the slope of the scale factor is, because we measure that from the Hubble Constant. And so, in an appropriate set of units, the scale factor is increasing at a rate of the Hubble Constant, which is, I think, 2 x 10-18 per second. So, if the scale factor is one right now, then one second from now, the scale factor is going to be 1 + 2 x 10-18. So, every second, it gets 2 x 10-18 bigger. That's not a real large increase and, in fact, over the entire history of humanity, it won't increase by that much. Let's see, in a million years, there is something like — okay, in 106 years, there are 3 x 107 x 106 seconds.

That's 3 x 1013 seconds. Multiply that by 2 x 10-18 and the scale factor increases by 3 x 1013 x 2 x 10-18.

That's 6 x 10-5. So, it still hasn't made a whole lot difference, even a million years from now. But, keep at it for a while, and the scale factor is going to pile up and will increase. So, we know that it's increasing.

And then, the question is, "Is this increase changing with time?" So, let's see. If it doesn't change — if it's just a straight linear increase, this is the situation in which there's no matter in the Universe. So, the Universe is not slowing down because of gravity, because there's no matter in it. And that gets labeled with this factor Omega [Ω] being equal to 0.

Let me remind you: Ω is defined as the density of the Universe divided by the critical density. The critical density is how dense it has to be to turn around. So, another possible track for the Universe: we know it's expanding at a particular rate now, but it could slow down in the future, turn around and fall back — could look something like this. This is the case where Ω is greater than 1. And then, the intermediate case where Ω is less than 1, but not equal to 0. It sort of does this. It falls off a little bit, continues to expand, something like that. And this Ω less than 1, but still greater than 0.

Okay. So, here's the goal. We've got to figure out which track we're on. At the moment, we sit here, and we measure the Hubble Constant. So, all we know is where we are and where we're going in the short run.

It's actually not a single number. It's actually a function. Right? It's a function of time. The scale factor changes with time. So, when I make this grandiose statement that I'm going to talk about the past and future of the Universe, what I really mean is, I want to determine what the scale factor is as a function of time.

And so, that's just a standard kind of a graph. Here's the scale factor, usually given the letter a. Function of t. Here's time. The present is somewhere along this graph. Call that now. And so, here's — and the Universe has some scale factor at the present time. And what we're going to do is we're going to call that — define the scale factor at the present time to be 1. We're just going to — that's called normalizing: when you set — when you multiply something by some convenient number so that it comes out to be 1 at a convenient moment. So, that's just a definition. We define the current scale factor to be 1, to be unity. And so, there's a point here at — right now, the scale factor is 1.

We also know what the slope of the scale factor is, because we measure that from the Hubble Constant. And so, in an appropriate set of units, the scale factor is increasing at a rate of the Hubble Constant, which is, I think, 2 x 10-18 per second. So, if the scale factor is one right now, then one second from now, the scale factor is going to be 1 + 2 x 10-18. So, every second, it gets 2 x 10-18 bigger. That's not a real large increase and, in fact, over the entire history of humanity, it won't increase by that much. Let's see, in a million years, there is something like — okay, in 106 years, there are 3 x 107 x 106 seconds.

That's 3 x 1013 seconds. Multiply that by 2 x 10-18 and the scale factor increases by 3 x 1013 x 2 x 10-18.

That's 6 x 10-5. So, it still hasn't made a whole lot difference, even a million years from now. But, keep at it for a while, and the scale factor is going to pile up and will increase. So, we know that it's increasing.

And then, the question is, "Is this increase changing with time?" So, let's see. If it doesn't change — if it's just a straight linear increase, this is the situation in which there's no matter in the Universe. So, the Universe is not slowing down because of gravity, because there's no matter in it. And that gets labeled with this factor Omega [Ω] being equal to 0.

Let me remind you: Ω is defined as the density of the Universe divided by the critical density. The critical density is how dense it has to be to turn around. So, another possible track for the Universe: we know it's expanding at a particular rate now, but it could slow down in the future, turn around and fall back — could look something like this. This is the case where Ω is greater than 1. And then, the intermediate case where Ω is less than 1, but not equal to 0. It sort of does this. It falls off a little bit, continues to expand, something like that. And this Ω less than 1, but still greater than 0.

Okay. So, here's the goal. We've got to figure out which track we're on. At the moment, we sit here, and we measure the Hubble Constant. So, all we know is where we are and where we're going in the short run.

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