sábado, 26 de novembro de 2011

16. Smooth Tension and Acceleration - Sean Carroll - Dark Matter, Dark Energy: The Dark Side of the Universe

This is the breakthrough lecture of the course. Sean Carroll is our expert guide and only now do we realize just how ideal the opportunity given us can be. Sean goes through this lecture quickly enough to require much rewinding for my review. In addition to the usually terse course guide, I believe this summary will help.

Sean admits up front that this lecture will attempt to translate equations into words. By nature this can be incomplete and less rigorous than the equations, but there is a need for the best translation possible. Typically after the initial vague description of dark energy, we move onwards and upwards without really grasping the concept. But there are no new concepts in this lecture, just Sean taking a time out to express some previous concepts more fully. We now realize that we shouldn't have actually understood before this point, and that we need more lectures like this one!

So far we've said dark energy is both smooth and persistent. This lecture describes why the persistence causes the expansion of the universe to accelerate. Two explanations are given, the first with negative pressure and the second without it. Sean admits to liking the second explanation better, but both have their own insights.

Dark Energy can be thought of as a negative pressure. This will be fully described soon. This contrasts with the energy density of the universe, which is positive. Gravity in Einstein's relativity responds to the addition of these two factors, whose sum could be negative or positive. The negative pressure of dark energy turns out to cause the total to be negative, thus causing the acceleration of the universe. This apparent anti-gravity affect seems to violate the classic conservation of energy. If dark energy is persistent regardless of time, and the universe is expanding over time, how can that total energy of the universe continue to increase without breaking some conservation law? That question will take some explaining:

Imagine an expanding box of matter and radiation. The total energy in the form of matter stays constant, but the total energy in the form of radiation decreases as the wavelengths are stretched. So the total energy of matter and radiation in the expanding box decreases. Our universe has persistent dark energy whose total increases as the universe expands, while the matter and radiation total energy decrease. There is just no reason for conservation of energy in relativity for an expanding universe.

This contrasts with Emmy Noether's Theorem where symmetry implies a conserved quantity. The concept of Time Translation Invariance are laws of physics and their playground that do not change with time. This symmetry implies a conservation of energy. General Relativity does allow a dynamical universe where the playground is not invariant. This break with symmetry allow energy to not be conserved.

To go even further, if we characterize those dynamics of the expanding universe, we can then know how the total energy changes. The example Sean uses is of a simple piston where the gas inside exerts a positive pressure. Pulling on the piston takes energy out of the gas and it works with the force of the hand. Pushing it puts energy into the gas which then works against the force of the hand. Photons in the universe also have this same positive pressure.

Negative pressure is a concept hard to wrap one's mind around, but here goes. Pulling on the piston actually puts energy into the gas from the hand, so it would work against the force of the hand, creating a tension. The gas exerts a constant force regardless, so with an increase in volume, the total energy increases inside the piston. This analogy fits with out description of dark energy a few paragraphs above.

So dark energy is a negative pressure, but that sounds a bit harsh. Why not smooth tension? It still shows dark energy is a misnomer regardless. Also, would pushing on the piston take energy out of the gas due to its decrease in volume, and work with the force of the hand? Does it matter since the universe is expanding not contracting? And what about the hand anyway? Is there an analogy to the hand in the real universe? Isn't the energy from the pulling and pushing hand conserved? Yes, but the universe has energy of its own that is not conserved.

We go back to the equations of Maxwell and Einstein to describe their unification. Electricity unifies with magnetism by Maxwell. Space unifies with time by Einstein's special relativity. Curvature unifies with energy and momentum by Einstein's general relativity, where gravity now includes energy in all of its forms such as pressure.

That pressure is equal to the energy density + (3 x pressure)

The factor of three is due to the three dimensions of space. This allows the negative pressure of dark energy to win over the positive pressure of the energy density, and thereby accelerate the expanding universe. But how can this jibe with the negative pressure of the piston where it worked against the force of the hand when pulling outwards and expanding the volume? Sean uses the analogy of air pressure in a room not directly affecting your hand because it pushes on all sides at once. But there is an indirect impact on gravity due to the curvature of spacetime which makes the universe expand faster and faster.

The second explanation is actually favored by the author and does not include negative pressure. Dark energy is persistent with expansion. So a constant impulse is given to the energy of the universe, causing acceleration.

Imagine a universe where the Friedmann equation has curvature (K) set to zero and the energy density (rho) is made up only of dark energy, which then equals the Hubble constant squared. So rho is also constant and proportional to Ho in this fake universe. But dark energy supplies the acceleration of the universe, so how can Ho be constant? It turns out Ho does not control the velocity of a galaxy, but the distance does. If Ho is constant or decreasing more slowly than distance is increasing, then you still get an acceleration. So there is no need for the concept of negative pressure in this model.

Energy contributes to the curvature of spacetime. In a flat universe, energy contributes to the actual expansion rate of spacetime. If that contribution does not go away, the expansion rate will persist.

But in the conventional universe, with rho made of matter and radiation, Ho was very high at the birth of the universe and low at the end. The contribution to the expansion rate does go away and the expansion rate will not persist.

Whew! What can I say. This type of lecture is what The Teaching Company is all about. Does it get any better than this?

is is a lecture Sean gives with only some trepidation. It's the lecture in which we don't learn anything new about the universe, about dark matter and dark energy. We just take something that we've already been saying about dark energy, and try to really understand it on a deeper level.

This is a lecture that you'll not ordinarily hear about in lectures on dark energy and the accelerating universe, since it's easier just to say some words that sound like they make sense, and then quickly move onto the next thing! So the purpose of this lecture is to go deeply into those words that sound like they make sense, then become convinced that they shouldn't have made sense in the first place. Yet if you think about it hard enough, they do begin to make sense again! So Sean is still not sure whether this is even the right thing to do, although it's valuable enough to begin to really understand what's going on when we start talking about dark energy.

What we've said about dark energy is that is has two very crucial properties, it's smoothly distributed through space, so the same amount of dark energy is right here in this cm³, as somewhere way in between galaxies and clusters very far away. At least the data are telling us that there's not substantially more dark energy inside the galaxies and clusters than in between them. That kind of makes sense, since had there been more dark energy inside a cluster, we would have noticed its affect on the gravitational field in the dynamics of the stuff inside the cluster.

The other part of dark energy is that it is persistent. The energy density(ρ) inside the dark energy is approximately constant as the universe expands. That's telling us that the dark energy is not made of some kind of particles that are becoming more and more dilute as the universe gets bigger. If that were the case, the energy would be going down. There would be fewer particles per cm³. Whatever the dark energy is, it stays the same as the universe gets bigger. So that's something we'll have to get into, namely the list of the possible candidates for stuff that could have a persistent energy density(ρ).

Yet instead, today we'll talk about the idea that if you have a persistent ρ, it makes the universe accelerate. Why is it that if the ρ of stuff doesn't go down, the manifestation of that in observable quantities is a universe that expands faster and faster? In fact, there are two different ways to explain this, which are just two different sets of words that we attach to the same set of equations. The truth is that this project of attaching words to equations is just because, as human beings, we like to have some intuitive grasp about what is going on.

The equations themselves are completely unambiguous. There's no question about what does happen. The equations are telling you absolutely once and for all, that if ρ does not change as the universe expands, it will make the universe accelerate. It's perfectly clear at the level of the equations themselves.

Yet we would like more than that. We'd like to go beyond just being able to write some equations down, to really get a deeper understanding of why it is like that. These are our attempts to attach words and concepts that make sense to us, onto those equations. Sometimes these attempts are going to necessarily be incomplete or fuzzy in some way. They will make a certain amount of sense to us, yet they are not nearly as rigorous as the equations themselves.

So we'll get two different explanations for why persistent dark energy makes the universe accelerate. Neither one is wrong, though we may find one explanation more compelling than the other one. That's perfectly OK, and is all up to us. The first explanation you'll often hear, is the following. We'll get the whole thing first, and then sort of unpack it and see if it makes sense.

The explanation says that dark energy has a negative pressure. In addition to positive ρ, there's also a pressure. Yet this pressure is a negative number. Now the thing that gravity responds to, according to Einstein, is a combination of ρ and pressure, which when added up, can be a negative number if the pressure is sufficiently negative. Even if you have a positive ρ, you can still get a negative gravitational effect, in the presence of negative pressure.

That's what happens when you have dark energy, and that's why it makes the things in the universe move apart faster and faster. It's almost like anti-gravity, pushing things apart. So we can decide for ourselves, whether or not this set of words actually made sense. So now we'll go into them more deeply and attempt to understand why our concept of dark energy would have a negative pressure, and why something with negative pressure would make the universe accelerate.

Before we get there, we have to understand a little bit about the very notion of conservation of energy. This is one of the most cherished concepts in physics, going all the way back to Galileo, if not earlier. We want to know how it works in the context of relativity, which is after all, a different theory than classical Newtonian mechanics. It turns out there actually isn't any necessary, logical reason, that once we understand the laws of physics better, it needs to continue to be the case that our old cherished notions are still true.

In the context of relativity, the way that the conservation of energy is manifested, is different. In fact, you could say, without being incorrect, that the energy in General Relativity is just not conserved. So now we'll try to make sense of that statement.

You may have noticed that the energy of the universe seems not to be conserved in the presence of dark energy. On the one hand, we've said that dark energy has an energy per cm³ that is approximately constant, and maybe even exactly constant. The dark energy is vacuum energy, so is a strictly, absolutely fixed, amount of energy per every cm³. On the other hand, the universe is getting bigger with expansion, so there are more and more cm³ in space as it expands. Therefor isn't it true that the total amount of energy in the universe is going up, and doesn't this mean that the energy is not conserved? The amount of energy is growing. So the answer is yes, the energy is growing, and it is not conserved.

Then you're supposed to say, "Isn't that bad, and violates our cherished notions of conservation of energy?" The answer is that it is not bad, that it actually makes perfect sense within the context of General Relativity. So to convince us of this, let's point out that even without dark energy, it is still the case that energy is not conserved in an expanding universe. Just think about an expanding universe that is more conventional. It has nothing in it but photons and ordinary matter particles, stuff that we certainly know exists.

So in a given region of the universe, which is expanding, there are a number of particles in that region which stays the same as the universe expands. Now there's two different regimes for the kinds of particles we can consider. There are matter particles which move slowly compared to the speed of light, and there are radiation particles that move at, or close to, the speed of light. So in each of these cases we get a different formula, a different idea of what the energy per particle is.

For a matter particle, it's E=mc². For a slowly moving particle, most of the energy of a particle is in its rest mass. This means that as the universe expands, the energy in each individual matter particle, stays constant. So because this region of space is expanding, the total number of particles in that region is constant, and the total energy in matter in that region, will remain fixed as the universe expands, just as you might hope it would do.

On the other hand, consider the energy of radiation in that same box. There, you have a fixed number of radiation particles, so like matter, that isn't changing. Yet the energy per particle is going down. The effect of the stretching of spacetime, is to increase the wavelength of every individual photon. The kinetic energy, if you like, of the radiation particles is diminishing because of the expansion of the universe. Therefor, if you add up the total amount of energy contained in radiation, in that box, as it expands, you don't get a constant. Yet get a number that decreases as the universe expands.

So there is actually an amusing psychological effect going on here. With dark energy, as the universe expands, the total energy is not conserved because it's going up, and that bothers people! With radiation, as the universe expands the total energy is not conserved because it's going down, yet this doesn't seem to bother people as much! From the equation point of view, both of those are exactly equally good or bad. If energy is conserved, the energy should not change, which means it should not go up or go down.

So the truth is that in an expanding universe, or in General Relativity more generally, there's no reason for the total energy of stuff in the universe to be conserved. By stuff we don't mean spacetime, but those substances that are in spacetime, whether dark energy, dark matter, ordinary matter, radiation, or what have you.

If you were to dig down into the classical laws of physics as Isaac Newton proposed them, there is a reason why energy is conserved. The deep reason why energy is conserved was actually first understood by a mathematician named Emmy Noether, who figured out something we call Noether's Theorem, which says that if there is a symmetry of nature, then associated with every symmetry is a conserved quantity. Something that is a symmetry of nature, implies there is some number you can calculate that never changes.

It doesn't quite go the other way. So just because something is a conserved quantity, doesn't mean it comes from a symmetry, but it is in fact, very often the case that it does. Energy for example, is conserved because of a certain symmetry of nature. What symmetry of nature is it? That would be time translation invariance. It's the statement that the laws of physics and the playground on which physics happens, don't change as time goes on. That is a true statement in Newtonian mechanics, where space and time are fixed and constant, where space does not change, so nothing happens to space, and the laws of physics also remain unchanged.

Therefor, in Newtonian mechanics you can derive and go through a set of equations that leads you to the conclusion that energy must be conserved. Yet General Relativity works differently, since it allows space and time to be dynamical. In particular we know that in cosmology the universe is expanding. In other words, the playground, the stage on which physics plays itself out, is not invariant under time. The universe of the past was different than that of the future. Therefor the deep reason we had to believe in energy conservation is no longer true. It is not, deep down, a surprise that in an expanding universe, energy is no longer conserved. Nevertheless, there is still an understanding of what happens. It's not that chaos has broken loose because the universe is expanding.

There used to be, in Newtonian mechanics, a rule that says the total energy is constant. Now in General Relativity there is a new rule, yet there's still a rule! This rule governs how the energy changes as the universe expands. If we can tell exactly how the universe is expanding, we can then tell exactly how the energy will change in response to that.

Fortunately there's actually a very easy way to understand this rule, in terms of a much more mundane system, namely a piston in exactly the same sense of the ones in our car, in the engine that is turning the energy in our gasoline, into the kinetic energy of our car's motion. In a piston, we imagine the simplest possible case in which we have a piston pushing into some substance, and we're changing the volume of that substance by pushing the piston in, or by pulling it out.

Now if you have an ordinary gaseous state of matter (not gasoline) inside the piston, such as air or anything like that, it takes energy to push it in. You can get energy out of it, by allowing the piston to come out. We say that the gas in the piston has a positive amount of pressure. The gas in the piston is pushing on the piston, and therefor if we just let it go, we could hook it up to a little engine. This is actually what you do inside your car, and you're getting energy out of the piston by allowing it to expand. That's what a positive pressure does, by saying you can extract energy by increasing the volume.

That's exactly what happens to a gas for example, made of photons. In the universe we can imagine trying to apply the phrase "gas" to a collection of photons bouncing off the walls of our little piston. Photons are going to bounce into the piston and exert a force on it, which is what we perceive as the pressure. So the photons in the piston will push on the piston, and we can extract energy by increasing the volume.

That's just what the universe does. The universe, by expanding, takes energy away from those photons. If instead, inside the piston, we have a bunch of particles that weren't moving, a bunch of particles that were motionless, like matter particles, then we wouldn't get any energy out by pulling on the piston. Again, that's exactly what happens in the universe. It expands and matter particles don't loose any energy in that way.

So positive pressure means that as we increase the volume, we take energy away. Therefor you might guess that what negative pressure means, is that as we increase the volume, we put energy in. That's what a negative pressure is. So if you try to imagine some physical system inside the piston that would have a negative pressure, it's something that when you pull on the piston, the system pulls back.

For example, we could imagine a complicated system of rubberbands or springs inside the piston, that were tied to its walls. If there's a rubberband going from one end of the piston to the part that we're pulling on, when we pull, it will then pull back on us. That's a negative pressure, or equivalently it's called a tension. A rubberband has tension, so that it takes energy to make it bigger. It's not giving us energy when we make it bigger.

So now let's think what it would be like for a piston to be full of dark energy? This is a special kind of non-physical thought experiment, where we imagine a piston with dark energy inside and no dark energy outside. So we just have zero energy everywhere outside. Inside we have a system with the property that the amount of energy in every cm³ is a constant. So what happens if we take that piston and try to pull on it, so that we are increasing the volume inside the piston? The energy per cm³ inside remains constant, so the total amount of energy inside the piston goes up.

In other words, we have to put energy into the system in order to pull out our piston. That is, if it makes sense, the proof that dark energy has a negative pressure. Dark energy is a system that requires energy for you to make it bigger. We need to put energy into the system somehow.

Since negative pressure is sometimes called tension, Sean has occasionally, semi-seriously argued, that dark energy is not a good name for dark energy! Everything in the universe has energy, and there's lots of things that are dark. So the essence of dark energy is not really correctly described by calling it dark energy. The important things about dark energy are that it is smoothly distributed, and that it has a negative pressure or a tension. So Sean proposes that we call dark energy smooth tension, which is both more accurate and kind of sexier than dark energy. It did not catch on though, which must mean Sean was just too late in coming up with this name!

So everything hangs together in this picture of how dark energy works. If you have an object or a physical system that has tension, that is negative pressure. Then when you expand the volume it takes up, you're putting energy into it. So contrary wise, if you have a system whose energy remains constant, you know that it has a negative pressure.

However, in the case of the piston, there was an external agent. There was an outside without any dark energy, and with someone pulling on it. There is no equivalent to this in the case of the actual universe. There is nothing outside of the universe, pulling on it or pushing on it. The universe is just evolving in accordance with Einstein's equations, so the analogy breaks down a little bit there.

If you counted the energy inside the piston, or the energy pulling on it, or pushing in it, that total energy would be conserved. As far as we know, according to our current theories anyway, there isn't anything outside, pushing or pulling on the universe. It's just that the universe has an energy of its own, that is not conserved. So we have to learn to deal with that.

Now let's see if that helps us to understand why the universe is accelerating. Let's grant that dark energy is associated with negative pressure. Dark energy is something that takes energy to make the volume bigger and bigger. So then, so what? Well, we have to go back to Einstein's equation, which tells us how the curvature of spacetime responds to stuff. Einstein's equation has a left-hand side involving the curvature of space and time, and has a right-hand side involving what we call the energy momentum tensor (vector) Tμν.

Rμν - ½Rgμν = (8πG)Tμν

In other words, Einstein's equation of General Relativity involves a unification of different concepts, just like Special Relativity does. Remember that Special Relativity was inspired by Maxwell's theory of electromagnetism, which unified our descriptions of electricity and of magnetism. Special Relativity itself unified our idea of space with our idea of time, into one notion of spacetime. General Relativity unifies the stuff that causes a gravitational field.

It used to be, according to Isaac Newton, that the stuff which caused gravity was mass. So in Special Relativity, Einstein says that mass is a form of energy, while in General Relativity he says the stuff that causes gravity is every form of energy. So it's not just mass that causes gravity, but it's also momentum, pressure, or strain, making a whole bunch of different ways in which energy can manifest itself.

So the thing that appeared on the right-hand side of Einstein's equation, the energy momentum tensor Tμν, is not just ρ, but it's also the pressure. So if you work through some math, which we can't quite do right now, you find that there's a rough guideline which says that in some cm³ of space, the thing that makes it expand or contract, according to gravity, is not just ρ, but a sum of ρ and pressure. That particular sum is as follows:

energy density(ρ) + 3(pressure)

Why is it a factor of three times the pressure? It's due to the three dimensions of space. If we lived in a universe with 5 dimensions of space, the force of gravity on a cm³, would be ρ + 5(pressure). So what that factor of 3 means, is that if you have a pressure that is equal but opposite to ρ, then the pressure wins out in the formula over ρ, since there is a factor of 3 multiplying the amount of the pressure.

If you have a ρ which is positive, and a pressure which is negative and comparable to ρ, then ρ + 3(pressure) will be a negative number. That means that instead of pulling space together, like you might guess, a sufficiently negative pressure pushes space apart and makes the universe accelerate. It's kind of like anti-gravity in the sense that things move apart from each other under the influence of gravity, rather than coming together.

So we could probably stop there and be willing to "buy" into this. Yet let's just point out a tiny little slight of hand that happened in the argument. It's not a lie that misleads us, but there's something that goes by very quickly that is worth paying attention to. The gravitational effect of the pressure is what is being talked about here. A negative pressure, remember, inside the piston, pulls on the piston. It wants to decrease the volume, because that saves energy. Yet what we're saying here is that negative pressure in the universe, makes it accelerate. How did that happen? What just went on?

What went on is, if there is pressure in every direction, nothing happens. It's like there being no force of the air pressure in the room on our hand, in one direction or the other, since the pressure is acting equally and oppositely on all sides of our hand. We don't feel any net force due to the pressure of the air in this room, even though the amount of pressure is some 15 pounds per square inch. If we only had that pressure on one side, it would be pushing our hand over quite strongly.

The same thing happens with the negative pressure of the dark energy in the universe. It's exactly the same at every point, in every direction, and therefor you feel precisely nothing from the direct impact of the negative pressure of the dark energy. On the other hand, there is a gravitational effect, an indirect influence of the negative pressure, due to its impact on the curvature of spacetime, and that affect is to make space expand faster and faster.

So all of those words are true. They do hang together and make sense, yet we need to "buy in" to the claims that dark energy has negative pressure, that there's a certain formula for the expansion of space (ρ + 3(pressure)), and the pressure itself exerts no net direct affect.

Therefor Sean can't help but resist giving us his own favorite explanation for exactly the same phenomenon. Why does a constant ρ make the universe accelerate? We'll now explain why, without referring to the concept of negative pressure at all. So here is the other explanation, which is equally good, yet we may like it better.

The explanation says that dark energy is persistent, so ρ does not go away but remains constant as the universe expands. Therefor, ρ gives a constant, persistent impulse, to the expansion of the universe. That persistent impulse in every cubic cm³ of space, manifests itself as acceleration.

That's more or less the set of words we said before, so let's unpack it a little bit more, to understand the deep meaning behind the chain of logic. We need to go back to the Friedmann equation, the equation in cosmology that relates the curvature of spacetime (K) to the energy density (ρ).

((8πG)/3)(ρ)=H² + K

Yet now to save ourselves a bit of the conceptual work, we'll set the spatial curvature (K) to zero. We have data from the CMB which verifies K is flat, or zero. So it's good enough to look at the Friedmann equation without the spatial curvature term:


In that case, we get a very simple relationship of ρ being proportional to H². That expansion rate H, of course, is measured by the Hubble constant, the Hubble parameter which relates the velocity that you observe in a galaxy, to the distance that it is from you:

v = Hd

So if the universe had nothing in it except for dark energy, just to do a simple thought experiment, what would happen? We would have constant ρ. If you have a constant ρ, and ρ is proportional to H², then H itself is also constant.

In conventional cosmology, when matter and radiation are important back during the early universe, the Hubble parameter (H) was much larger than today. Yet in this fake, toy universe we're discussing just for the moment, with no spatial curvature, no radiation, not anything but dark energy, you would have a Hubble parameter that was truly constant and never changed as the universe expanded.

So we're allowed to ask, "Wait a minute, we just said that the dark energy made the universe accelerate? How can it be that a universe where the Hubble parameter is constant, is accelerating? In an accelerating universe, shouldn't the expansion rate be going up? The answer is no, and this is just one of those miracles of non-Euclidean geometry which manifests itself in General Relativity.

A constant expansion rate corresponds to an accelerating universe, because the expansion rate is not a velocity. The Hubble constant (H) is not giving you directly the velocity of a galaxy, but it's telling us that if you know the distance to a galaxy, what would the velocity be? We return to the Hubble law:

v = Hd

This very simple equation says that the velocity of a distant galaxy is the Hubble constant, times the distance. So H relates all sorts of different galaxies to the velocities you observe. Now ask what happens according to Hubble's law, if H is strictly constant and not changing? Just look at one galaxy and then let the universe expand. The velocity you observe, is the distance to that galaxy times the Hubble constant. Yet as the universe expands, that distance is increasing. Therefor, if the Hubble constant doesn't change, the velocity that we perceive will also increase.

That's what we really mean, when we say that the universe is accelerating. We say that if you look at one galaxy, and follow its velocity as a function of time, that velocity goes up. The galaxy moves away from you faster and faster. That's what it means to live in an accelerating universe.

In a decelerating universe, in one that only had matter and radiation in it, the distance would still increase. Yet the Hubble parameter would decrease even faster. That's why you would see the velocity of a distant galaxy decrease in a decelerating universe.

The statement that the universe is accelerating, is equivalent to the statement that the Hubble parameter is either constant, or decreasing more slowly than the distance is increasing. That's what an accelerating universe really means.

Let's say exactly the same thing in a different set of words. One way of thinking about what the Hubble parameter is telling us, is how long it takes for the universe to increase in size by some fixed number. So let's say the Hubble parameter is telling us that it takes 10 billion years for the universe to double in size. Let's furthermore say that the Hubble parameter is constant. This is more or less what we have in our current universe.

So that's saying that every ten billion years, the universe doubles in size. What that means is, if we take two galaxies that are 1 billion light years apart, and you wait 10 billion years, they will then be 2 billion light years apart. Wait another 10 billion years, and they will be 4 billion light years apart. Anther 10 billion, they'll be 8 billion apart, etc.

It's just "they told two friends," and "they told two friends." These galaxies move apart at an apparent velocity which is ever increasing. The reason why, is that every cm³ of space is expanding at a constant rate, a constant expansion rate of space. Plus, an increasing amount of space in between the galaxies leads to an acceleration of the two galaxies away from each other.

That is Sean's favorite explanation for why dark energy makes the universe accelerate. We don't need to go through the intermediary of a negative pressure after all, which is a kind of difficult idea to grasp all by itself. Instead, you can just say that what dark energy does, is contribute to the curvature of spacetime.

In a flat universe, that means dark energy contributes to the expansion rate of spacetime. If that contribution doesn't go away, then the expansion rate of spacetime will persist. If we lived in a universe without dark energy, in one with only ordinary matter and radiation, the far future of the universe is one in which it expands evermore slowly. It gets emptier and emptier, and approaches exactly the situation you would have if there was no stuff in the universe, if you were not expanding whatsoever. It would be a static, unexpanding, empty spacetime.

Yet with dark energy, with stuff that doesn't go away, there's always something that is making the universe expand with this constant magnitude. The manifestation of that to us, is that we see individual galaxies moving away faster and faster.

So hopefully, all this now makes perfect intuitive sense to us, except we're still left with the question of what this dark energy actually is? So in the next lecture, Sean will be wearing a different tie, and we'll start thinking seriously about different candidates for just what the dark energy might be.

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