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Your Daily Equation #30: What Sparked the Big Bang?

Join us for #YourDailyEquation with Brian Greene. Every Mon – Fri at 3pm EDT, Brian Greene will offer brief and breezy discussions of pivotal equations. Even if your math is a bit rusty, tune in for accessible and exciting stories of nature and numbers that will allow you to see the universe in a new way.

Episode 30 #YourDailyEquation: Even after astronomical observations convinced Einstein that the universe is expanding there remained the question of what drove the universe to expand in the first place–that is, what sparked the Big Bang. Join Brian Greene to explore one prominent possibility: repulsive gravity ignited a brief but powerful burst off cosmic inflation.

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Speaker 1:

Hey everyone, welcome to this next episode of, yes, you know what it is, Your Daily Equation. And today, I’m going to take up the question of what drove the Big Bang? What banged? What is responsible for the outward swelling of space that we currently witness? Now, let me put this in a little bit of historical context, covering some ground that I have discussed in some previous episodes, but I like these conversations, one-way conversations, though it’s not one way. I look at your comments. I do. And sometimes you see that I respond to them. So I like these conversations to be as self-contained as they can be. Can’t be fully self-contained, but let me just give a little bit of background that I’ve covered before.

As you know, and as we’ve seen, back in around the 1920s or so, Alexander Friedmann, actually let me bring them up here. Alexander Friedmann, Georges Lemaître, they used Einstein’s newly minted description of the force of gravity, his general theory of relativity, to come to a conclusion about the universe. And to simplify slightly, they both pretty much recognized that the universe should be swelling. It should be expanding. Einstein himself resisted this idea until the astronomical observations, not that he made, this might suggest that Einstein was actually at the telescope. This is a visit that Einstein took, I think, to Mount Wilson Observatory with Edwin Hubble. So Hubble’s observations of the distant galaxies showed that they are all racing away.

And at that point, Einstein changed his perspective from the older view that the universe was fixed, eternal, unchanging on the largest of scales, and took on this perspective that the universe is expanding. I think it was in this visit, not to digress, that Einstein and Elsa, his wife/cousin, they were visiting Mount Wilson Observatory, and Elsa got to talking to some of the astronomers there and asked them what they do. And they described how they have all this fancy equipment to understand the universe, its properties and its history and so forth. And Elsa apparently responded by saying, “Oh, my husband does that on the back of an envelope.”

So yeah, that’s the kind of support that you need, absolutely. So anyway, so this is the situation. 1929, Einstein agrees with the observations, and in fact, with the prediction of his own equations, that the universe should be expanding. Sorry. Now, this leaves open an important, and that’s the question for today, which is, okay, the universe is expanding, but what drove the expansion to start? What sparked the Big Bang? There must’ve been some kind of outward pushing force that started everything rushing out, space and time themselves as well, swelling. What force could that be? Now, Einstein’s own equations are basically silent on this question. Because if you take Einstein’s math and run it all the way back to time zero, use the math to wind that cosmic film back to the beginning, sorry, I have some phlegm today, the equations break down.

By break down, it’s basically you get things equivalent to dividing by zero. You know that that, in any calculator, computer, whatever, dividing by zero doesn’t make sense. And you get that kind of a so-called singularity in Einstein’s equations. Bringing everything here today. So that’s the issue. Einstein’s math itself doesn’t give you the answer. So this was puzzling, and people thought about this for a long time. I’m now going to describe one potential answer. It’s not the only answer for the history of the universe leading up to the form that we witness today, but certainly is a version of the story that many people take quite seriously and have taken seriously for many decades. And that is so-called inflationary cosmology. So let me just switch over to my iPad. Good. So what we’re talking about, as I said is, what banged? What drove the Big Bang? And the answer that I’m going to focus attention on is this idea of cosmic inflation.

And I should say that the originators of this idea, Alan Guth started it all off, Andrei Linde, important, vital contributions, Paul Steinhardt and Andreas Albrecht as well, very important contributions. Not all of them agree on whether this is the right way of talking about cosmology today. At the very end, I’ll mention how certainly Paul Steinhardt has become a critic of the very approach that he was instrumental in founding. But that’s kind of the beauty of science, right? People don’t walk in lock step. It’s not that there is some kind of pronouncement from on high, the priests of physics declare how the universe was. No, people look out at the data. They look at the equations and sometimes even their very perspective that they had at one moment in time can change. And that’s the beauty of the way we follow the zigzagging path toward truth when we’re talking about the kind of truth that science and physics in particular can reveal.

Okay, so what is cosmic inflation? And the idea is pretty straightforward. It makes use of the critical equation of cosmology that I discussed in a previous episode. But again, to be self-contained, I will remind you of that equation. So if you define something called the scale factor, a(t), and what the scale factor is, it tells you the distance between two objects, let’s say two galaxies. We often use galaxies as the tracers of the size of space. So imagine that we have a couple of galaxies, completely arbitrary shapes. But imagine that at some initial time they are a distance … I’m going to write this a little bit differently. Imagine at some initial time they’re at distance d0 apart. The idea is that the scale factor, which can talk about the overall stretching of space, it stretches all distances by the factor a(t).

So at time t then, the separation between the two galaxies will be a(t) times d0. Why do we call it the scale factor? Again, you could imagine that this is all sitting inside a kind of schematic map of space, a map of the universe. And then on any map, there is this thing called the legend, where you indicate how to go from distances on the map to real distances in the real world. And imagine that there’s an a(t) in that legend, where that is a scaling factor which says, “Hey, one inch or one centimeter may have been one light year,” if it’s a big map, at initial time, ti, but at a later time, you now are going to multiply that scaling by a(t). So now one centimeter or one inch [inaudible 00:08:19] a(t) times one light year. So it’s basically this kind of stretching of the distances given by how a(t), say, grows, if it’s growing in time.

Now, what we want to have, therefore, is an equation for a(t) to know how it changes in time. And this is what I discussed in an earlier episode. I’m just going to write down the equation that emerges from Einstein’s field equations of the general theory of relativity. And there are a couple of equations that are relevant, but the one that really will be most potent for us today is this one, d2a/dt squared, divided by a equals -4 pi G times … Now, I’m going to … 4 pi G over 3 times rho plus 3p. It’s really 3p over C squared. Then I’m going to set C equal to one. So rho here is the energy density. This guy over here, p, is the pressure.

And so if you know the material that’s within a region of space and you’re able to calculate how much energy it has, how much pressure it exerts, that allows you to write down the right hand side of this equation. And then you just have this fairly simple differential equation to solve. And this will give you the scale factor, say, as a function of time. Now, one thing. Somebody asked me about this. I should quickly say it. Now, a here again is scale factor. There’s something slightly confusing because this term over here, d2a dt squared, that’s how the scale factor is accelerating, like d2x dt squared is acceleration, but a itself is not acceleration, a is the scale factor. So just don’t get those two confused. I’m sure most of you didn’t.

All right, so this is the key equation. What do we do with this? Well, the essential point is this, if the right hand side of this equation, namely rho plus 3p, if that happened to be negative, if that happened to be negative, then that negative sign together with this negative sign, we deal with a positive sign on the right hand side, which would mean that the acceleration of the scale factor would be positive. And that’s the kind of thing we’re looking for, right? If you’re looking for a bang, you kind of want to start with some small size. You want it to rapidly swell. So you’re going from the scale factor, say not growing, perhaps initially, that’s one possibility, and then it’s rapidly growing. So that’s a rapid acceleration, and that’s the kind of thing that you want. Now, you can have that in this theory because although energy density, this guy, is positive, as I discussed in a previous episode, pressure can be negative.

And negative pressure simply means pressure that’s not pushing outward like gas in a balloon or gas in a room, but rather pressure that pulls inward, sort of like a stretched rubber sheet or a stretched rubber band wants to pull inward. There’s sort of tension there. So any substance that does that is something that can source on the right-hand side of this equation, an amount of rho plus 3p that will drive, say, a rapid swelling of space and accelerating change in the scale factor. Okay, then, so what’s the idea? Well, I noted in, again, a previous episode, again, just to be complete here that Einstein’s own cosmological constant is one example of a thing, a substance, or at least a quality of space, however you want to think about it, in which rho plus 3p is negative, because for it, it turns out … I haven’t explained this part. And at some point I will. You can’t do everything in a short period of time. But p equals minus rho for this cosmological constant.

This is therefore a constant energy density, spread throughout space, whose pressure is minus that energy density. And then if you put p equals minus rho in the expression rho plus 3p, of course, then you do get a negative number. In this particular case, it could say -2 rho. And that does drive an outward swelling. Now, the problem with a cosmological constant is it’s constant. For a Big Bang, we don’t want a constant bang, if you will. We want it to just be something that lasts for a short period of time, makes the universe big, and then we live through the aftermath of that initial, rapid, rapid swelling. The cosmological constant is constant. It doesn’t do that. So we want something that depends upon time. And the answer for that substance that might play that role comes from the physicist I mentioned before. And it makes use of a subject that we at some point may talk about, quantum field theory. But it uses the most basic qualities of quantum field theory, so it’s not something that will take us too far afield for me to quickly describe.

What is a field? Well, let me just give it a name. Conventionally, we usually call the field phi. It’s usually in this setup called the inflaton field. You are familiar with other fields, of course, like the electromagnetic field. What is that? That is a quality of space, if you will, that at every location, the electromagnetic field in space gives you a magnitude and a direction for the electric field and for the magnetic field. Now, the field that we’re talking about here is a so-called scalar field, so it doesn’t have direction. It just has a value at every location in space. The analogy that people often use is temperature. You can talk about the temperature at every location in space. It can slowly vary, of course, from location to location. A field is kind of like that.

It is a substance that has a magnitude for a scale of field at every location in space. And you can write down the properties of this field. You can write down what its energy looks like and what its pressure looks like. Let me just record those equations. So the energy associated with the field, well, it has parts that depend on how the field is varying in space, how the field is varying in time, and also the potential energy that the field has. Maybe I should just draw a little picture of that first. Oftentimes, we have some potential energy function. Please write. Thank you very much.

So if this is the value of the field, and this is the potential, there are various shapes that you can draw, something like this. And that tells you if the value of the field is given by this number over here, I’ll call phi 0, then the potential energy is V (phi 0). And there’s no God-given … Maybe that’s not the right word. There’s no way that we currently know of to determine the potential of a field from first principles. We choose the potential curve for a field in order that it performs whatever function we need it to perform in our theory. So you should bear that in the back of your mind.

But the point is, if the field has a potential function given by V, then its energy comes from say three sources, how the field changes in time, if a field is rapidly oscillating, it has more energy than if it’s not rapidly oscillating, plus how the field changes in space. If the field is varying from location to location, there’s an energy associated with the desire, in some sense, of the field to have a constant value. So gradients, changes in the field’s value across space, contributes energy, as well as this potential energy function. And what I’m going to do is I’m going to assume that we have a field whose value across space is constant. So I’m not going to consider this contribution here, but I will allow it to change in time. Because as I mentioned, that’s the whole point of not using a cosmological constant. We want the field’s value to change in time so that I can source, perhaps, a rapid swelling of space in the early universe, and then it shuts off.

So how would we do that? So let me write it down. So the form typically is (d phi/dt) squared plus V(phi). Now this first term, 1/2 (d phi/dt) squared, that should kind of makes sense, because if you think of phi as x, then the xdt is velocity. So (d phi/dt) is like the velocity of the field. It’s how quickly the field’s value’s changing. In classical mechanics, the kinetic energy is proportional to 1/2 v squared, 1/2 mv squared. So we have 1/2 v sub phi squared, and that therefore, the first term makes sense. The second term I’ve already described. Good. What about the pressure? If you’re talking about pressure, how does that go? By the way, this is really energy density. I’m talking about energy, but you’d have to integrate this over all space.

So this is really this guy that we called rho. Cool. Well, say cool. I use the E for energy, but it’s really energy in a given volume of space. That’s what I mean by calling that rho. What about pressure? We can also write down an equation for pressure. It’s a little more involved to derive that equation. You get it from the stress energy [inaudible 00:19:01] T mu nu on the right-hand side of Einstein’s equations. I’m just going to write down the answer for that one, which is very similar. It’s 1/2 (d phi/dt) squared minus V(phi). That’s the only change, the plus to the minus. Now notice the following. If d phi/dt is very small, and everything’s relative here, so it’s small compared to the value of V(phi), of course, then you will note that rho is very close to V(phi).

P is very close to -V(phi), and therefore P is very close to -rho. That’s good. P close to -rho means that in this equation, over here, we’re in a very similar situation to what we had over here, which is a net positive on the right-hand side of this equation. And that, therefore, can source a rapid increase in the scale factor, d2a/dt squared positive. This is actually exponential growth when you realize that you have an a here in the bottom. If you solve the differential equation d2a/dt squared equals some positive number times a, then you know that this would go like e to the square root of that number, t, times some value, say, at time zero. You take the first derivative, it brings down one square root, second derivative brings down a second, and therefore you solve the differential equation.

So you’ve got exponential growth if you have a nearly constant value on the right-hand side of this equation. So that then is the basic idea. We just have to set things up so that the rate of change of the field with respect to time is really small. How do you do that? Well, here’s the idea that historically first was written down by Linde and also by Steinhardt and Albrecht. Imagine that you have a potential energy curve that has the following shape. It’s really kind of flat for a while. And then it kind of gradually rolls down to this value here. So this is the value of phi. Now, if you think about the value of phi as if it’s like a little ball, when the ball is on this hill, it will slowly roll in this direction, but it won’t have a high speed. So d phi/dt will not be big because the gradient of the potential is so small.

And then finally, it will roll down more quickly, which means that in this region over here, you will have this exponential growth in the scale factor a, and again, that’s just what we had right over here. The right-hand side of the equation is nearly constant because d phi/dt is nearly equal to zero. It’s non-zero, but it’s much smaller than the value of the potential. And the value of the potential here is not changing, hardly changing at all. It’s got such a small slope to it. So you get nice exponential growth in that period. And then in this section over here, inflation, which this is called, I guess I didn’t use that language, this inflationary burst from d2a/dt squared being positive, inflation is coming to an end in this region, because d phi/dt is now becoming larger and therefore this relation, p equal -rho, is being violated.

And if you plug this into the right-hand side of this equation, ultimately, this will not be negative, and therefore you will not have a negative times a negative giving you the positive, which is the rapid growth. The accelerated increase in the value of the scale factor. So then, what is this idea? First of all, this is called slow roll inflation. It’s just one example of inflation. But the idea is the bang in the Big Bang is coming from right in here, this nearly constant value of the inflaton field in, say, some region of space, nearly constant across that region, nearly constant time, although it’s slowly decreasing. That sources, this repulsive gravity, that’s what this is again. This is repulsive gravity coming from the fact that in this equation over here, this minus sign, together with this being negative, yields a net positive, and that gives this guy rapidly growing, outward pushing. So the outward pushing force is nothing but the repulsive gravity sourced by nearly constant value of this inflaton field in a region of space, across a small interval of time.

So here’s where the bang happens, and then the field rolls down. And frankly, I won’t talk about it here, but when it rolls down, some people write a potential like this, allowing this ball to slosh up and down as it comes to rest. And as it sloshes up and down, it can actually produce particles. That’s where particles come from, the energy inflaton field. It rolls down, slosh, slosh, slosh. As it sloshes, its energy is drained by the production of particles that ultimately are the particles that make up you and me and everything else.

So that’s this basic idea of this version of inflationary cosmology. It gives you a sense of where the bang in the Big Bang comes from. And let me just show you a quick picture then. I don’t know, just to make it a little bit dramatic. So when the inflaton field is in this region over here, here’s what happens. You get this rapid swelling of space, this rapid increase in the scale factor, ultimately rolls, down allowing particles to form, that ultimately come together under the more usual attractive gravity, yielding stars, galaxies, and all the other structure in the universe. Now, can you test this idea? Well, kind of. What do I mean by that? Well, here’s the idea. What I just described to you was a completely classical picture, right? Balls rolling down hills. That’s a very classical notion. You need to overlay quantum mechanics on this picture. And if you do that, you realize that when the field is rolling down the hill, its value can actually slightly fluctuate at different locations, not enough to ruin the analysis, but it overlays slightly different values of the inflaton field at different locations.

That then translates into slightly different values at the energy density, slightly different values of the duration that inflation will last. And when you take all of that into account, it gives rise to slightly different values of the temperature across space that we should measure now. This is the so-called temperature variations, anisotropies, in the cosmic microwave background radiation, slightly different temperatures coming from the quantum overlay of jitters in the inflaton field as this process takes place. And then inflation kind of stretches these little tiny, tiny quantum fluctuations. It stretches them across the sky. So here is a picture of the temperature variations across the sky. The different colors represent really tiny temperature variations on the order of, say, one part a hundred thousand. So it’s kind of a tour to force to be able to measure these temperature variations. But here’s my point.

The point is you can do a statistical analysis on how these temperature values vary across the sky. And then you can compare it with a calculation which overlays quantum effects on everything I just told you today to get a prediction for how the temperature should vary if these ideas are correct.I’m now going to show you a little graph that shows you the mathematical prediction from quantum effects overlaid on the inflationary story that I just told, you together with the measurements. And here it goes. That solid curve, that’s the result of mathematical calculations using inflationary cosmology and quantum mechanics, and the dots here are the measured value of the temperature. And look at that. Look at the spectacular agreement between the measured value of the temperature variations and the predicted one coming out from this inflationary story. And look, that’s remarkable, right?

We measure the temperature variations through very sensitive telescopes that capture photons that have been traveling toward us since about 380,000 years after the bang. By measuring those photons, we can measure the temperature of the region from which they emanated. And the slightly different values of the temperature that we get give rise to the dots on that curve, and the dots agree with the mathematical calculation done by these little creatures crawling around on planet Earth, making use of some mathematical equations. I mean, wow, it’s kind of mind blowing. So you might, at that point say, “Done. We now have proof, slam dunk proof that we understand the bang in the Big Bang.” That would probably … Well, certainly in my mind and in many people’s minds, but not everybody. I think it’s too quick. I am definitely one who has confidence in this approach, but I don’t think it’s been proven.

There’s vigorous argument. If you feel that I’m hedging my conversation right now, it is. I’m not exactly sure how far into this controversy I want to go, but I’ll talk for a couple of minutes, just to finish up this episode. There is a controversy happening right now between those who think that inflationary cosmology or some version of inflationary cosmology is it, we understand at least … not everything, but we understand a great deal about what happened in the early universe. And I should perhaps say, how long … Oh, let me just bring my iPad back up on the screen here. Sorry, I should have said this before. How long is the duration of time over which the value of the inflaton field is slowly changing so that the inflationary expansion is happening? And it probably depends. It completely depends, really, on exactly the shape of the potential energy curve that we write down.

But as a rule of thumb, let’s say the expansion happens, I don’t know, 10 to the -33, -32, -34 seconds. The point is, it’s a really tiny fraction of a second. And in that tiny fraction of a second, the power of the repulsive push is so enormous that the universe can expand to larger than the entire visible universe. So this is a truly bang. I should have said that. I mean, this is a rapid swelling of space. I showed you in the imagery, but just so you see where it comes from in the mathematics. But I bring this point up here because so much of the conclusions about what inflationary cosmology tells you is dependent on what things were actually like in the early universe. For instance, we are assuming that there was an inflaton field. We’re assuming that it had a potential of a particular shape. We are therefore making some critical assumptions about the initial conditions in the early universe.

And that’s the main point. People do not agree on what the initial conditions of the early universe were. And until we truly understand the initial conditions, the stuff that was there and the potential of any curves, if that’s relevant in whatever theory that we’re talking about, there’s going to be significant controversy. And the controversy goes a little bit further in inflationary cosmology because of one final point, which I’ll now describe. As a universe rapidly swells, the inflaton field continues to fill that region during the inflationary expansion. And that means there’s more and more inflaton field, if you will. In fact, as the inflaton field is going under its quantum fluctuations, its quantum jitters … Let me just come back here for half a second. I thought this was going to be that short episode, but you know me.

So as it’s doing its quantum jitters, sometimes it’s going to jitter a little down, but sometimes it actually can jitter a little up. What does that mean? That means that it’s virtually impossible to ever fully use the inflaton field up. So one region of space rapidly swells, for instance, but there’s left over, if you will, inflaton field in the overall environment. It’s still filling space. In some regions, it’s going to continue to roll down, but in some it’ll actually jitter up and then jitter down a little bit and jitter up and so forth. So it doesn’t simply all go away. And that means you can source other big bangs, other rapidly swelling … They will continue to undergo the rapid swelling, even if an our region, ultimately the inflaton field rolls down. And that gives rise to what’s known as the inflationary multiverse. Just sort of have a picture of it in your mind. Here goes one version.

So the inflaton field in our little region sources our swelling, but the swelling continues. In some sense, you got bang after bang after bang, giving rise to universe after universe after universe. And when you do the analysis, you find that in those other universes, say, the cosmic microwave background radiation that would arise from the quantum jitters of the inflaton field in that universe could look very different from the picture that I showed you. And yet, the inflationary theory would be able to explain that different pattern of temperature variations across the night sky in that universe. So that’s a little disturbing because it kind of says that it’s hard to prove inflation wrong, because had the temperature variations been different here, we could have explained it by saying, “Well, we’re just in one of those universes where the quantum fluctuations just give rise to a different spectrum of anisotropies.”

Now, the nice thing is that we don’t have to stand on our head carefully adjusting the shape of the potential energy curve to explain our observations in this universe. And that sense of Occam’s razor, the simplicity of the potentials that at least can come pretty close to observations as those observations get more and more refined, we have to adjust things more and more, suggest to some that inflation is just a non-falsifiable framework. And this is a point that’s been made by Paul Steinhardt, leading him to introduce other approaches to cosmology, the bouncing cosmology that he and his collaborators have put forward. And again, as I said at the outset, I think it’s great that there are competing approaches. My own view is that inflation offers this wonderful package, that in a very simple mathematical formalism is able to relatively easily describe observations that we do make here in this universe.

I’d be more comfortable if we could deal more fully with this multiverse aspect of the theory. Not all versions of inflation yield a multiverse. Some of them look unnatural in order to avoid the multiverse quality. So on and on the conversation will go. This should just give you a flavor that it’s an ongoing debate as to the status of the inflationary theory within the notion of fundamental descriptions of the early universe. And that’s something that hopefully observations may one day settle.

For instance, inflationary cosmology naturally gives rise to oscillations in the fabric of space-time, primordial, gravitational waves. If we were to detect those, that would be one more piece of evidence in the inflationary column. And that would be a wonderful development. We thought we had this some years ago. It turned out that the observations were being misinterpreted, but that’s the nature of the game. But the point of this episode of Your Daily Equation, of course, is just to show you how very simple mathematical equations can give rise to a burst of repulsive gravity in the early universe, and that burst of repulsive gravity is potentially the bang in the Big Bang. All right, that’s it for today. We will take this up in our next episode, or take up something else or take up something related. Until then, take care.

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Your Daily Equation #30: What Sparked the Big Bang?

Join us for #YourDailyEquation with Brian Greene. Every Mon – Fri at 3pm EDT, Brian Greene will offer brief and breezy discussions of pivotal equations. Even if your math is a bit rusty, tune in for accessible and exciting stories of nature and numbers that will allow you to see the universe in a new way.

Episode 30 #YourDailyEquation: Even after astronomical observations convinced Einstein that the universe is expanding there remained the question of what drove the universe to expand in the first place–that is, what sparked the Big Bang. Join Brian Greene to explore one prominent possibility: repulsive gravity ignited a brief but powerful burst off cosmic inflation.

Transcription

Speaker 1:

Hey everyone, welcome to this next episode of, yes, you know what it is, Your Daily Equation. And today, I’m going to take up the question of what drove the Big Bang? What banged? What is responsible for the outward swelling of space that we currently witness? Now, let me put this in a little bit of historical context, covering some ground that I have discussed in some previous episodes, but I like these conversations, one-way conversations, though it’s not one way. I look at your comments. I do. And sometimes you see that I respond to them. So I like these conversations to be as self-contained as they can be. Can’t be fully self-contained, but let me just give a little bit of background that I’ve covered before.

As you know, and as we’ve seen, back in around the 1920s or so, Alexander Friedmann, actually let me bring them up here. Alexander Friedmann, Georges Lemaître, they used Einstein’s newly minted description of the force of gravity, his general theory of relativity, to come to a conclusion about the universe. And to simplify slightly, they both pretty much recognized that the universe should be swelling. It should be expanding. Einstein himself resisted this idea until the astronomical observations, not that he made, this might suggest that Einstein was actually at the telescope. This is a visit that Einstein took, I think, to Mount Wilson Observatory with Edwin Hubble. So Hubble’s observations of the distant galaxies showed that they are all racing away.

And at that point, Einstein changed his perspective from the older view that the universe was fixed, eternal, unchanging on the largest of scales, and took on this perspective that the universe is expanding. I think it was in this visit, not to digress, that Einstein and Elsa, his wife/cousin, they were visiting Mount Wilson Observatory, and Elsa got to talking to some of the astronomers there and asked them what they do. And they described how they have all this fancy equipment to understand the universe, its properties and its history and so forth. And Elsa apparently responded by saying, “Oh, my husband does that on the back of an envelope.”

So yeah, that’s the kind of support that you need, absolutely. So anyway, so this is the situation. 1929, Einstein agrees with the observations, and in fact, with the prediction of his own equations, that the universe should be expanding. Sorry. Now, this leaves open an important, and that’s the question for today, which is, okay, the universe is expanding, but what drove the expansion to start? What sparked the Big Bang? There must’ve been some kind of outward pushing force that started everything rushing out, space and time themselves as well, swelling. What force could that be? Now, Einstein’s own equations are basically silent on this question. Because if you take Einstein’s math and run it all the way back to time zero, use the math to wind that cosmic film back to the beginning, sorry, I have some phlegm today, the equations break down.

By break down, it’s basically you get things equivalent to dividing by zero. You know that that, in any calculator, computer, whatever, dividing by zero doesn’t make sense. And you get that kind of a so-called singularity in Einstein’s equations. Bringing everything here today. So that’s the issue. Einstein’s math itself doesn’t give you the answer. So this was puzzling, and people thought about this for a long time. I’m now going to describe one potential answer. It’s not the only answer for the history of the universe leading up to the form that we witness today, but certainly is a version of the story that many people take quite seriously and have taken seriously for many decades. And that is so-called inflationary cosmology. So let me just switch over to my iPad. Good. So what we’re talking about, as I said is, what banged? What drove the Big Bang? And the answer that I’m going to focus attention on is this idea of cosmic inflation.

And I should say that the originators of this idea, Alan Guth started it all off, Andrei Linde, important, vital contributions, Paul Steinhardt and Andreas Albrecht as well, very important contributions. Not all of them agree on whether this is the right way of talking about cosmology today. At the very end, I’ll mention how certainly Paul Steinhardt has become a critic of the very approach that he was instrumental in founding. But that’s kind of the beauty of science, right? People don’t walk in lock step. It’s not that there is some kind of pronouncement from on high, the priests of physics declare how the universe was. No, people look out at the data. They look at the equations and sometimes even their very perspective that they had at one moment in time can change. And that’s the beauty of the way we follow the zigzagging path toward truth when we’re talking about the kind of truth that science and physics in particular can reveal.

Okay, so what is cosmic inflation? And the idea is pretty straightforward. It makes use of the critical equation of cosmology that I discussed in a previous episode. But again, to be self-contained, I will remind you of that equation. So if you define something called the scale factor, a(t), and what the scale factor is, it tells you the distance between two objects, let’s say two galaxies. We often use galaxies as the tracers of the size of space. So imagine that we have a couple of galaxies, completely arbitrary shapes. But imagine that at some initial time they are a distance … I’m going to write this a little bit differently. Imagine at some initial time they’re at distance d0 apart. The idea is that the scale factor, which can talk about the overall stretching of space, it stretches all distances by the factor a(t).

So at time t then, the separation between the two galaxies will be a(t) times d0. Why do we call it the scale factor? Again, you could imagine that this is all sitting inside a kind of schematic map of space, a map of the universe. And then on any map, there is this thing called the legend, where you indicate how to go from distances on the map to real distances in the real world. And imagine that there’s an a(t) in that legend, where that is a scaling factor which says, “Hey, one inch or one centimeter may have been one light year,” if it’s a big map, at initial time, ti, but at a later time, you now are going to multiply that scaling by a(t). So now one centimeter or one inch [inaudible 00:08:19] a(t) times one light year. So it’s basically this kind of stretching of the distances given by how a(t), say, grows, if it’s growing in time.

Now, what we want to have, therefore, is an equation for a(t) to know how it changes in time. And this is what I discussed in an earlier episode. I’m just going to write down the equation that emerges from Einstein’s field equations of the general theory of relativity. And there are a couple of equations that are relevant, but the one that really will be most potent for us today is this one, d2a/dt squared, divided by a equals -4 pi G times … Now, I’m going to … 4 pi G over 3 times rho plus 3p. It’s really 3p over C squared. Then I’m going to set C equal to one. So rho here is the energy density. This guy over here, p, is the pressure.

And so if you know the material that’s within a region of space and you’re able to calculate how much energy it has, how much pressure it exerts, that allows you to write down the right hand side of this equation. And then you just have this fairly simple differential equation to solve. And this will give you the scale factor, say, as a function of time. Now, one thing. Somebody asked me about this. I should quickly say it. Now, a here again is scale factor. There’s something slightly confusing because this term over here, d2a dt squared, that’s how the scale factor is accelerating, like d2x dt squared is acceleration, but a itself is not acceleration, a is the scale factor. So just don’t get those two confused. I’m sure most of you didn’t.

All right, so this is the key equation. What do we do with this? Well, the essential point is this, if the right hand side of this equation, namely rho plus 3p, if that happened to be negative, if that happened to be negative, then that negative sign together with this negative sign, we deal with a positive sign on the right hand side, which would mean that the acceleration of the scale factor would be positive. And that’s the kind of thing we’re looking for, right? If you’re looking for a bang, you kind of want to start with some small size. You want it to rapidly swell. So you’re going from the scale factor, say not growing, perhaps initially, that’s one possibility, and then it’s rapidly growing. So that’s a rapid acceleration, and that’s the kind of thing that you want. Now, you can have that in this theory because although energy density, this guy, is positive, as I discussed in a previous episode, pressure can be negative.

And negative pressure simply means pressure that’s not pushing outward like gas in a balloon or gas in a room, but rather pressure that pulls inward, sort of like a stretched rubber sheet or a stretched rubber band wants to pull inward. There’s sort of tension there. So any substance that does that is something that can source on the right-hand side of this equation, an amount of rho plus 3p that will drive, say, a rapid swelling of space and accelerating change in the scale factor. Okay, then, so what’s the idea? Well, I noted in, again, a previous episode, again, just to be complete here that Einstein’s own cosmological constant is one example of a thing, a substance, or at least a quality of space, however you want to think about it, in which rho plus 3p is negative, because for it, it turns out … I haven’t explained this part. And at some point I will. You can’t do everything in a short period of time. But p equals minus rho for this cosmological constant.

This is therefore a constant energy density, spread throughout space, whose pressure is minus that energy density. And then if you put p equals minus rho in the expression rho plus 3p, of course, then you do get a negative number. In this particular case, it could say -2 rho. And that does drive an outward swelling. Now, the problem with a cosmological constant is it’s constant. For a Big Bang, we don’t want a constant bang, if you will. We want it to just be something that lasts for a short period of time, makes the universe big, and then we live through the aftermath of that initial, rapid, rapid swelling. The cosmological constant is constant. It doesn’t do that. So we want something that depends upon time. And the answer for that substance that might play that role comes from the physicist I mentioned before. And it makes use of a subject that we at some point may talk about, quantum field theory. But it uses the most basic qualities of quantum field theory, so it’s not something that will take us too far afield for me to quickly describe.

What is a field? Well, let me just give it a name. Conventionally, we usually call the field phi. It’s usually in this setup called the inflaton field. You are familiar with other fields, of course, like the electromagnetic field. What is that? That is a quality of space, if you will, that at every location, the electromagnetic field in space gives you a magnitude and a direction for the electric field and for the magnetic field. Now, the field that we’re talking about here is a so-called scalar field, so it doesn’t have direction. It just has a value at every location in space. The analogy that people often use is temperature. You can talk about the temperature at every location in space. It can slowly vary, of course, from location to location. A field is kind of like that.

It is a substance that has a magnitude for a scale of field at every location in space. And you can write down the properties of this field. You can write down what its energy looks like and what its pressure looks like. Let me just record those equations. So the energy associated with the field, well, it has parts that depend on how the field is varying in space, how the field is varying in time, and also the potential energy that the field has. Maybe I should just draw a little picture of that first. Oftentimes, we have some potential energy function. Please write. Thank you very much.

So if this is the value of the field, and this is the potential, there are various shapes that you can draw, something like this. And that tells you if the value of the field is given by this number over here, I’ll call phi 0, then the potential energy is V (phi 0). And there’s no God-given … Maybe that’s not the right word. There’s no way that we currently know of to determine the potential of a field from first principles. We choose the potential curve for a field in order that it performs whatever function we need it to perform in our theory. So you should bear that in the back of your mind.

But the point is, if the field has a potential function given by V, then its energy comes from say three sources, how the field changes in time, if a field is rapidly oscillating, it has more energy than if it’s not rapidly oscillating, plus how the field changes in space. If the field is varying from location to location, there’s an energy associated with the desire, in some sense, of the field to have a constant value. So gradients, changes in the field’s value across space, contributes energy, as well as this potential energy function. And what I’m going to do is I’m going to assume that we have a field whose value across space is constant. So I’m not going to consider this contribution here, but I will allow it to change in time. Because as I mentioned, that’s the whole point of not using a cosmological constant. We want the field’s value to change in time so that I can source, perhaps, a rapid swelling of space in the early universe, and then it shuts off.

So how would we do that? So let me write it down. So the form typically is (d phi/dt) squared plus V(phi). Now this first term, 1/2 (d phi/dt) squared, that should kind of makes sense, because if you think of phi as x, then the xdt is velocity. So (d phi/dt) is like the velocity of the field. It’s how quickly the field’s value’s changing. In classical mechanics, the kinetic energy is proportional to 1/2 v squared, 1/2 mv squared. So we have 1/2 v sub phi squared, and that therefore, the first term makes sense. The second term I’ve already described. Good. What about the pressure? If you’re talking about pressure, how does that go? By the way, this is really energy density. I’m talking about energy, but you’d have to integrate this over all space.

So this is really this guy that we called rho. Cool. Well, say cool. I use the E for energy, but it’s really energy in a given volume of space. That’s what I mean by calling that rho. What about pressure? We can also write down an equation for pressure. It’s a little more involved to derive that equation. You get it from the stress energy [inaudible 00:19:01] T mu nu on the right-hand side of Einstein’s equations. I’m just going to write down the answer for that one, which is very similar. It’s 1/2 (d phi/dt) squared minus V(phi). That’s the only change, the plus to the minus. Now notice the following. If d phi/dt is very small, and everything’s relative here, so it’s small compared to the value of V(phi), of course, then you will note that rho is very close to V(phi).

P is very close to -V(phi), and therefore P is very close to -rho. That’s good. P close to -rho means that in this equation, over here, we’re in a very similar situation to what we had over here, which is a net positive on the right-hand side of this equation. And that, therefore, can source a rapid increase in the scale factor, d2a/dt squared positive. This is actually exponential growth when you realize that you have an a here in the bottom. If you solve the differential equation d2a/dt squared equals some positive number times a, then you know that this would go like e to the square root of that number, t, times some value, say, at time zero. You take the first derivative, it brings down one square root, second derivative brings down a second, and therefore you solve the differential equation.

So you’ve got exponential growth if you have a nearly constant value on the right-hand side of this equation. So that then is the basic idea. We just have to set things up so that the rate of change of the field with respect to time is really small. How do you do that? Well, here’s the idea that historically first was written down by Linde and also by Steinhardt and Albrecht. Imagine that you have a potential energy curve that has the following shape. It’s really kind of flat for a while. And then it kind of gradually rolls down to this value here. So this is the value of phi. Now, if you think about the value of phi as if it’s like a little ball, when the ball is on this hill, it will slowly roll in this direction, but it won’t have a high speed. So d phi/dt will not be big because the gradient of the potential is so small.

And then finally, it will roll down more quickly, which means that in this region over here, you will have this exponential growth in the scale factor a, and again, that’s just what we had right over here. The right-hand side of the equation is nearly constant because d phi/dt is nearly equal to zero. It’s non-zero, but it’s much smaller than the value of the potential. And the value of the potential here is not changing, hardly changing at all. It’s got such a small slope to it. So you get nice exponential growth in that period. And then in this section over here, inflation, which this is called, I guess I didn’t use that language, this inflationary burst from d2a/dt squared being positive, inflation is coming to an end in this region, because d phi/dt is now becoming larger and therefore this relation, p equal -rho, is being violated.

And if you plug this into the right-hand side of this equation, ultimately, this will not be negative, and therefore you will not have a negative times a negative giving you the positive, which is the rapid growth. The accelerated increase in the value of the scale factor. So then, what is this idea? First of all, this is called slow roll inflation. It’s just one example of inflation. But the idea is the bang in the Big Bang is coming from right in here, this nearly constant value of the inflaton field in, say, some region of space, nearly constant across that region, nearly constant time, although it’s slowly decreasing. That sources, this repulsive gravity, that’s what this is again. This is repulsive gravity coming from the fact that in this equation over here, this minus sign, together with this being negative, yields a net positive, and that gives this guy rapidly growing, outward pushing. So the outward pushing force is nothing but the repulsive gravity sourced by nearly constant value of this inflaton field in a region of space, across a small interval of time.

So here’s where the bang happens, and then the field rolls down. And frankly, I won’t talk about it here, but when it rolls down, some people write a potential like this, allowing this ball to slosh up and down as it comes to rest. And as it sloshes up and down, it can actually produce particles. That’s where particles come from, the energy inflaton field. It rolls down, slosh, slosh, slosh. As it sloshes, its energy is drained by the production of particles that ultimately are the particles that make up you and me and everything else.

So that’s this basic idea of this version of inflationary cosmology. It gives you a sense of where the bang in the Big Bang comes from. And let me just show you a quick picture then. I don’t know, just to make it a little bit dramatic. So when the inflaton field is in this region over here, here’s what happens. You get this rapid swelling of space, this rapid increase in the scale factor, ultimately rolls, down allowing particles to form, that ultimately come together under the more usual attractive gravity, yielding stars, galaxies, and all the other structure in the universe. Now, can you test this idea? Well, kind of. What do I mean by that? Well, here’s the idea. What I just described to you was a completely classical picture, right? Balls rolling down hills. That’s a very classical notion. You need to overlay quantum mechanics on this picture. And if you do that, you realize that when the field is rolling down the hill, its value can actually slightly fluctuate at different locations, not enough to ruin the analysis, but it overlays slightly different values of the inflaton field at different locations.

That then translates into slightly different values at the energy density, slightly different values of the duration that inflation will last. And when you take all of that into account, it gives rise to slightly different values of the temperature across space that we should measure now. This is the so-called temperature variations, anisotropies, in the cosmic microwave background radiation, slightly different temperatures coming from the quantum overlay of jitters in the inflaton field as this process takes place. And then inflation kind of stretches these little tiny, tiny quantum fluctuations. It stretches them across the sky. So here is a picture of the temperature variations across the sky. The different colors represent really tiny temperature variations on the order of, say, one part a hundred thousand. So it’s kind of a tour to force to be able to measure these temperature variations. But here’s my point.

The point is you can do a statistical analysis on how these temperature values vary across the sky. And then you can compare it with a calculation which overlays quantum effects on everything I just told you today to get a prediction for how the temperature should vary if these ideas are correct.I’m now going to show you a little graph that shows you the mathematical prediction from quantum effects overlaid on the inflationary story that I just told, you together with the measurements. And here it goes. That solid curve, that’s the result of mathematical calculations using inflationary cosmology and quantum mechanics, and the dots here are the measured value of the temperature. And look at that. Look at the spectacular agreement between the measured value of the temperature variations and the predicted one coming out from this inflationary story. And look, that’s remarkable, right?

We measure the temperature variations through very sensitive telescopes that capture photons that have been traveling toward us since about 380,000 years after the bang. By measuring those photons, we can measure the temperature of the region from which they emanated. And the slightly different values of the temperature that we get give rise to the dots on that curve, and the dots agree with the mathematical calculation done by these little creatures crawling around on planet Earth, making use of some mathematical equations. I mean, wow, it’s kind of mind blowing. So you might, at that point say, “Done. We now have proof, slam dunk proof that we understand the bang in the Big Bang.” That would probably … Well, certainly in my mind and in many people’s minds, but not everybody. I think it’s too quick. I am definitely one who has confidence in this approach, but I don’t think it’s been proven.

There’s vigorous argument. If you feel that I’m hedging my conversation right now, it is. I’m not exactly sure how far into this controversy I want to go, but I’ll talk for a couple of minutes, just to finish up this episode. There is a controversy happening right now between those who think that inflationary cosmology or some version of inflationary cosmology is it, we understand at least … not everything, but we understand a great deal about what happened in the early universe. And I should perhaps say, how long … Oh, let me just bring my iPad back up on the screen here. Sorry, I should have said this before. How long is the duration of time over which the value of the inflaton field is slowly changing so that the inflationary expansion is happening? And it probably depends. It completely depends, really, on exactly the shape of the potential energy curve that we write down.

But as a rule of thumb, let’s say the expansion happens, I don’t know, 10 to the -33, -32, -34 seconds. The point is, it’s a really tiny fraction of a second. And in that tiny fraction of a second, the power of the repulsive push is so enormous that the universe can expand to larger than the entire visible universe. So this is a truly bang. I should have said that. I mean, this is a rapid swelling of space. I showed you in the imagery, but just so you see where it comes from in the mathematics. But I bring this point up here because so much of the conclusions about what inflationary cosmology tells you is dependent on what things were actually like in the early universe. For instance, we are assuming that there was an inflaton field. We’re assuming that it had a potential of a particular shape. We are therefore making some critical assumptions about the initial conditions in the early universe.

And that’s the main point. People do not agree on what the initial conditions of the early universe were. And until we truly understand the initial conditions, the stuff that was there and the potential of any curves, if that’s relevant in whatever theory that we’re talking about, there’s going to be significant controversy. And the controversy goes a little bit further in inflationary cosmology because of one final point, which I’ll now describe. As a universe rapidly swells, the inflaton field continues to fill that region during the inflationary expansion. And that means there’s more and more inflaton field, if you will. In fact, as the inflaton field is going under its quantum fluctuations, its quantum jitters … Let me just come back here for half a second. I thought this was going to be that short episode, but you know me.

So as it’s doing its quantum jitters, sometimes it’s going to jitter a little down, but sometimes it actually can jitter a little up. What does that mean? That means that it’s virtually impossible to ever fully use the inflaton field up. So one region of space rapidly swells, for instance, but there’s left over, if you will, inflaton field in the overall environment. It’s still filling space. In some regions, it’s going to continue to roll down, but in some it’ll actually jitter up and then jitter down a little bit and jitter up and so forth. So it doesn’t simply all go away. And that means you can source other big bangs, other rapidly swelling … They will continue to undergo the rapid swelling, even if an our region, ultimately the inflaton field rolls down. And that gives rise to what’s known as the inflationary multiverse. Just sort of have a picture of it in your mind. Here goes one version.

So the inflaton field in our little region sources our swelling, but the swelling continues. In some sense, you got bang after bang after bang, giving rise to universe after universe after universe. And when you do the analysis, you find that in those other universes, say, the cosmic microwave background radiation that would arise from the quantum jitters of the inflaton field in that universe could look very different from the picture that I showed you. And yet, the inflationary theory would be able to explain that different pattern of temperature variations across the night sky in that universe. So that’s a little disturbing because it kind of says that it’s hard to prove inflation wrong, because had the temperature variations been different here, we could have explained it by saying, “Well, we’re just in one of those universes where the quantum fluctuations just give rise to a different spectrum of anisotropies.”

Now, the nice thing is that we don’t have to stand on our head carefully adjusting the shape of the potential energy curve to explain our observations in this universe. And that sense of Occam’s razor, the simplicity of the potentials that at least can come pretty close to observations as those observations get more and more refined, we have to adjust things more and more, suggest to some that inflation is just a non-falsifiable framework. And this is a point that’s been made by Paul Steinhardt, leading him to introduce other approaches to cosmology, the bouncing cosmology that he and his collaborators have put forward. And again, as I said at the outset, I think it’s great that there are competing approaches. My own view is that inflation offers this wonderful package, that in a very simple mathematical formalism is able to relatively easily describe observations that we do make here in this universe.

I’d be more comfortable if we could deal more fully with this multiverse aspect of the theory. Not all versions of inflation yield a multiverse. Some of them look unnatural in order to avoid the multiverse quality. So on and on the conversation will go. This should just give you a flavor that it’s an ongoing debate as to the status of the inflationary theory within the notion of fundamental descriptions of the early universe. And that’s something that hopefully observations may one day settle.

For instance, inflationary cosmology naturally gives rise to oscillations in the fabric of space-time, primordial, gravitational waves. If we were to detect those, that would be one more piece of evidence in the inflationary column. And that would be a wonderful development. We thought we had this some years ago. It turned out that the observations were being misinterpreted, but that’s the nature of the game. But the point of this episode of Your Daily Equation, of course, is just to show you how very simple mathematical equations can give rise to a burst of repulsive gravity in the early universe, and that burst of repulsive gravity is potentially the bang in the Big Bang. All right, that’s it for today. We will take this up in our next episode, or take up something else or take up something related. Until then, take care.