This lecture speculates on one of the candidates for explaining dark energy, the vacuum energy. While viewing other new Teaching Company courses today, Anselm's Proof of God was discussed. Though astronomy is my background, I wonder if these two lectures are that different. Definitions and chains of arguments which I have no reason to know whether are true or not, which some leaders in the field dismiss outright as blasphemous, or which are neat to think about but could eventually turn into some long forgotten old theory. But humans are a curious lot, so we will think about them by nature.

General relativity and quantum theory have no reason for vacuum energy to equal zero. The energy of spacetime cannot be lowered below this value, whatever it is. But a change in this energy is most important, not necessarily its initial energy. Like the ball in a high school inclined plane demo, when the potential energy of the ball changes into kinetic energy. Raising the demo ten feet higher increases the potential energy, but the change from potential to kinetic is the same.

So with the total energy of the vacuum energy. General Relativity predicted gravity as manifested by all forms of energy. We could detect energy by means of gravity, such as precession of Mercury's perihelion. Vacuum energy also changes, but it is far too small to detect individually, but only through accumulated effects over cosmological distances.

Einstein predicted spacetime was the same everywhere around 1916-1918, but it was either expanding or contracting. Rather than some personal philosophy, Einstein thought the universe to be static by talks with his colleagues and being more than ten years before Hubble's discovery. As typical, Einstein changed his equations many times, so added a constant to preserve a static universe, even if physically implausible. But Hubble's 1929 announcement, and interpretations from Gammow, revealed Einstein to think this constant was the blunder of a lifetime. He was scooped by Hubble!

This Cosmological Constant, is our vacuum energy, which is only a candidate theory remember! Its value could be anything, with contributions from classical and quantum physics. But the value turns out to be much much larger than what is observed. Quantum physics has the inherent uncertainty and probability of course. Like a motionless pendulum, there will be some motion even at rest since position and momentum are not simultaneously knowable. And like the inclined plane, a change in height does not change the laws of physics or motion of the ball.

So too for the minimum amount of energy of spacetime in quantum fields. This is also true classically and is equal to a random number. But the quantum fields have a jiggle associated with them, of virtual particles. They carry energy and contribute to the vacuum energy. The quantum field tracks the number of particles, like bookkeeping, where the vacuum state is the lowest possible with no particles. But these virtual particles pop in and out of existence everywhere at all times and we can even detect their affects as they contribute to the energy density of empty space.

There are two contributions to this; an arbitrary classical number of which we have no idea, and the quantum mechanical fluctuations which we can estimate to be the same basic size as the vacuum energy. Theory predicts 10 to the 112th power ergs per cm³, while observations show 10 to the -8th power! This Cosmological Constant Problem was known well before the acceleration of the universe. Not nearly as large a difference, but due to some cancellation effect, predicted by symmetry, would take care of it.

But now dark energy contributes to the energy density of space. It's as if something made the vacuum energy disappear, like flipping a coin and getting heads for many many times but then a tail shows up! The vacuum energy either shouldn't be there or should be much larger. We are stuck with it, but it is there and we are at a loss of explaining it. Perhaps a better formula will solve it?

The dark energy equals the energy of empty space which fits the data, but we have no understanding of what it is. So we try other things of course. Why is the vacuum energy so small? Is the dark energy dynamic? That is up next.

We have every reason to be proud of what we've learned about the universe so far, both over the course of these lectures, and over the course of the past 100 years as working physicists and cosmologists. We know that the total ρ of the universe is about the critical density needed to make the universe spatially flat. We know ρ is about 5% ordinary matter, 25% dark matter, and 70% dark energy. We have a great theory for what the ordinary matter does. We have the standard model of particle physics, which tells us what the particles are, and how they interact, consistent with all the experiments we've done here on earth so far.

We don't know what the dark matter is, yet we have more than enough different candidates. It's not a surprise or deep mystery to us, how you could get dark matter to be 25% of the universe. We have lots of different ideas for what it could be, and even lots of good ideas about how to test those theories as well. This includes ways we could create the dark matter in particles individually, and also to detect it coming from outer space, here to our detectors on earth.

So now is the lecture in which we face up to the fact that dark energy is not so simple, that it's something about which we don't have very good ideas on, as far as composition. So we're going to spend three lectures looking at different possible theories for why the universe is accelerating and spatially flat, and for why we seem to believe there is dark energy. In the end, we'll realize that none of these theories are even especially promising. None of them are like the example of supersymmetry in the case of dark matter, where you have a theory that does other things, that naturally provides you with a candidate, and that whether right or not, is at least a very plausible scenario.

So let's think about exactly why we think that there is dark energy. We have a set of evidence for it, which is very different for the evidence we have for dark matter. The dark matter evidence comes from the local dynamics and behavior of stuff in the universe, attracting other stuff. We have galaxies, clusters of galaxies, gravitational lensing, the growth of structure in the universe, all saying there's more stuff here in those bound systems, than there is outside. If we naturally attribute that to cold, non-interacting, massive particles, we seem to fit the data.

Dark energy, on the other hand, is found globally. We look at the acceleration of the universe due to the kind of stuff that is inside it. We look at the spatial geometry of the universe, which is a way of measuring everything all at once. We find that if we imagine that 70% of ρ is smoothly spread throughout space, taking the same amounts inside clusters and galaxies, as well as outside of them, and also imagine ρ being persistent as a function of time, not redshifting or diluting away as the universe expands, then we can fit all these data, all at once. So we call this mysterious new substance dark energy.

So what could it be? The two things we know about dark energy are that it's smoothly distributed through space, and nearly constant in time. Therefor, certainly the simplest thing it could possibly be, is something which is absolutely 100% the same from place to place in space, as well as the same from moment to moment in time.

The data are not yet telling us that this is precisely the case. They are, of course, consistent with some slop in that conclusion. There could be some small variation from place to place. So that's going to be the subject of the next lecture, while this one will be about the possibility that it really is truly constant, and the physical underpinnings for those two possibilities are very different, even though they're observational signatures are pretty much similar.

So we should say before we go there, that even though we'll contemplate in this lecture the possibility that empty space is an absolutely constant energy, known as the vacuum energy, it's still only a candidate for what the dark energy could be. These are not synonyms with each other.

The dark energy is the label given to whatever it is that is out there in the universe, at a smoothly distributed and persistent 70%. The vacuum energy is a specific idea for what it might be, the energy of empty space itself. It's still possible that what we call the vacuum energy is zero, and what we call the dark energy that is making the universe accelerate, is something else entirely.

Yet for this lecture, we'll concentrate on the possibility of vacuum energy. So what do we actually mean by this term? We mean the energy of empty space itself. Yet what does that mean? It means that you take a little region of space, a little cm³, and you remove from it, everything you possibly can, so that it's completely empty. You remove all the ordinary matter, all the dark matter, all the radiation, all the neutrinos, so there's literally nothing there. You then ask how much energy is there, contained inside this cm³?

In the context of either General Relativity, our best theory of gravity, or of quantum field theory, our best theory of microscopic physics, there's no reason why the answer to that question should be zero. There is some number, some constant of nature, which tells us how much energy there is, in every region of empty spacetime.

So this picture that we see once again, is an artist's impression, Sean's impression, of what dark energy, vacuum energy, is like. You can think of it as a false color image of empty space itself. Of course, empty space is really completely dark and invisible, completely see-through and transparent, so this is just an attempt to make you contemplate the idea that even in this little bit of empty space, there is still some energy there.

The energy is not contained in some substance that is located there and could be moved around, but rather it's inherent in the fabric of space and time itself. That is the idea of vacuum energy, the bare minimum amount of energy you can fit anywhere. There's nothing you could do to that cm³, no physical process, which would lower the energy below that.

So what do we mean by the "energy of empty space?" We're saying over and over again, these same words, the energy that you have in every empty cm³ of space. Yet you're still sitting there thinking, "What does that really mean?" To a physicist, when you ask that question, you're asking, "What does it do, how would we know that it is there, is there some operational, observable consequence of saying these words?"

Now we get a little bit of a subtlety actually, because in almost all areas of physics, the actual value of the energy doesn't matter. What matters is how much energy changes when one process happens. The energy that goes up or goes down, or the energy that gets exchanged between two different ways in which it can manifest itself. For example, you might be familiar with high school or college physics experiments, such a ball or box rolling down an inclined plane. You'll talk about the energy contained in the ball, so it has potential energy depending on how high it is, and kinetic energy depending on how its moving, and if everything is very smooth, there is zero change in the total energy, so you're just converting total energy into potential energy.

Yet here's the thing. If you ask what the total amount of energy is, of that ball rolling down the plane, the answer doesn't actually even matter! If you did exactly the same experiment from the plane being right at a certain place with the ball rolling down, or if you raised the plane up so it's at a different place, then the potential energy of the ball would indeed change. Yet the way in which the total energy would change in going from potential energy to kinetic energy, wouldn't change. Also of course, the motion of the ball going down the plane would be exactly the same, whether at one place or the other. As long as the tilt of the plane is the same, the local physics is exactly identical, so the total amount of energy the ball has, is completely irrelevant.

So you might say that even if there were empty space energy, vacuum energy at every point in spacetime, how would we know? The answer is that gravity is the one thing in nature that really does care about the absolute amount of energy. Remember, Einstein tells us that gravity responds to everything, it's universal. So it will turn out to be the case that the amount of energy contained in empty space, doesn't affect anything about, for instance, the standard model of particle physics, but the only thing it affects is gravity. It acts to curve spacetime, and create a gravitational field. That's how you can detect it. That's how, in fact, we're claiming that we did, in fact, detect something like it, by looking at the curvature of space, and of course the acceleration of the universe.

So what is the effect of this vacuum energy? We've already said that it makes the universe accelerate. If there is some ρ in empty space, that ρ doesn't go away as the universe expands. It's a constant of nature from place to place, and time to time. So as the universe is expanding, the vacuum energy is giving a perpetual impulse to the expansion of space, and we perceive that as the acceleration of the universe.

Yet there are other affects. If you had infinitely good measuring apparatuses, you would be able to detect the existence of the vacuum energy, using all sorts of gravitational experiments. The classic experiment that lets us test General Relativity in the first place, is the motion of the perihelion of Mercury. The planet Mercury moves around the sun in an ellipse, and according to Isaac Newton, that ellipse should be fixed in orientation once and for all. If you put aside perturbations from the other planets, and imagine a perfect solar system that just has the sun and Mercury, that ellipse is exactly the same for all time.

Einstein comes along in General Relativity and says that this ellipse should change slightly. Of course, that was actually known to be the case already. Before Einstein actually came up with his theory, astronomers had already determined that Mercury was slowly shifting in its orbit. That was one way we were able to tell the General Relativity was a better theory than Newtonian gravity. It turns out that if you have a vacuum energy, you would add a small, new effect to the motion of the planet Mercury.

However, when we say small, that effect is really really small. There's no possible way we'll ever be able to test the possibility of vacuum energy by observing Mercury. It's just swamped with things like even our sneezing here on earth. so there's a million larger effects than vacuum energy, when you're thinking about how Mercury moves around the sun.

Yet the reason why we can detect the vacuum energy in cosmology, is that the effects accumulate. When you're looking at the motion of a galaxy or a supernova in some galaxy, there are many cm³ between us and that supernova. Each and every one of those cm³ is expanding just a little bit faster, because of the presence of vacuum energy. It's that accumulated effect over cosmological distances, that enables us to actually detect the fact that the vacuum energy is there.

The first way that scientists started thinking about this concept of vacuum energy, actually goes back to Einstein himself. Soon after inventing General Relativity, he started thinking about cosmology. He knew he had a very different way of thinking about space and time than Newton did, the picture of which both space and time were fixed and absolute. Einstein's picture had both space and time as a geometry, so they're dynamical and can change. So he started thinking about the entire universe.

So we're now talking 1916-1918, a decade before we knew the universe was expanding. Einstein took on as a simplifying assumption, that the universe was pretty much the same everywhere. Yet he found that within this possibility of the universe being pretty much the same everywhere, that his equation governing the curvature of space and time, was not finding a solution to which the universe was static. One in which it was smooth everywhere and more or less staying the same as time went on.

Now we know today the reason why this was so. The universe is in fact expanding, it's not static. That makes perfect sense to us, since if you have galaxies spread throughout the universe, they're going to pull on each other. In the context of Newtonian gravity, if you try to ask the question of what could happen if you filled space full of galaxies, with the same number everywhere, would they pull on each other or not? It's very hard to get an answer to this.

On the one hand you think that yes, they're pulling on each other, so yes they should come together. Yet there's an equally plausible analysis that says, "Here I am, this galaxy, with every other galaxy pulling on me, but they're equally distributed, so the net force is zero and I don't move." So nothing moves.

Within General Relativity there's a different answer that says you can think about nothing moving, but space itself can get bigger or smaller. So in General Relativity, if you have nothing but matter in the universe, the answer is unambiguous. Space better either be expanding, or contracting.

Now there's a little bit of a false story that goes around about Einstein, thinking about exactly these issues. It goes that Einstein was blinded by his philosophical presuppositions. He wanted to believe, this false story goes, that the universe is static, due to some religious or philosophical reasons. Therefor, when he found his equations didn't describe a static universe, he was upset and tried to change them.

The truth is that he did believe the universe was static, but it wasn't due to the philosophical predisposition. It's because he asked his astronomer friends, who at that time of 1915-1929, as far as anyone new, thought the universe was static because that's what the observations were telling us. It wasn't until 1929, when Hubble discovered the relationship between the velocity of distant galaxies and their distance, that we realized that space is in fact, expanding and dynamical.

So Einstein was just trying to fit the data. He thought the universe was static, yet realized that his own equations did not allow for a static solution if you just imagine a universe full of galaxies or matter of any sort. As a result, in 1917 he changed his equation of General Relativity by adding a new term which he called the cosmological constant.

Now these days, we consider changing Einstein's equations to be an interesting step, but a dramatic one, since we have such good success with them. Yet we have to remember that Einstein was coming up with new equations all the time. In fact his final equation was not anywhere near the first version, since he was changing things to fit both philosophical and experimental all along the way.

So finally by 1917 he said he could add a term the the let hand side of his equation. We see it again, with the left side telling us how curved space and time are, and the right-hand side representing stuff in the universe, energy, momentum, and so forth.

Rμν - ½Rgμν = (8πG)Tμν

He then put a new term on the left, for which he could then find a solution that the universe could be filled with galaxies, and yet not expanding or contracting. The solution he called the cosmological constant, we now know can be moved to the right hand side, and it's exactly the same as vacuum energy. in fact some people like to argue whether or not this term deserves to be on the left hand side with the spatial and spacetime curvature bits, or deserves to be on the right hand side with the energy bits. The truth is that it absolutely makes no difference whether a term in an equation is on the left or right, since it functions in exactly the same way, and has the same interpretation. Einstein called it the cosmological constant, and we will call it the vacuum energy, but it is the same thing.

So vacuum energy acts to make the universe accelerate, basically pushing things apart as the universe expands. Ordinary matter works to pull things together, so basically all Einstein did was to find a solution where those two effects exactly balance. Now it's true that you could find a solution where the universe is not expanding or contracting, but of course his solution was unstable. If you perturb it just a little bit, so that things start pulling together just slightly, that force would win and it would collapse. Yet if things start pushing apart slightly, that pushing apart force would win, and things would start accelerating. So Einstein found a solution that fit the data, but wasn't a physically plausible solution. It wasn't something that was robust to small changes in what happened.

Then of course, by 1929, Hubble comes along and says, "Well, the universe is not static, it's expanding just as Einstein could have predicted in 1917." According to George Gamow, one of the primary thinkers on the Big Bang model, Einstein later claimed that his biggest blunder as a scientist, was to introduce the cosmological constant into his equations. If he had not done that, he would have made a prediction. He would would have been able to say that, "Despite what the current observations say, circa 1917, I predict that the universe should be either expanding or contracting."

Yet he blinked and wasn't able to do that. He then said, "Away with the cosmological constant, and I'm sorry I ever invented it." But once you do that, you can't undo it. Pandora's box is not so easily closed. These days, we realize that what Einstein called the cosmological constant, is what we call the vacuum energy, and we can start asking questions like, "If there can be energy density in empty space, how much should there be? Can we make some sort of reasonable estimate for how much energy there should be in every cm³ of empty space? The minimum possible value?"

Well it's a very fuzzy story that goes back and forth. There is no precise answer to this question of what the vacuum energy should be, according to our current theories. The true answer is that it can be anything at all, it's a constant of nature. It's like asking, "What is the mass of the electron?" There is no ahead of time answer to what it should be, you have to go out there and measure it.

On the other hand, there are reasonable answers and unreasonable answers. We can do a pretty good job of estimating the order or magnitude for how large the vacuum energy should be. We have both, what we'll call a classical contribution to the vacuum energy, the thing that it should be if you ignored all of quantum mechanics, and then as we'll explain, quantum mechanics adds new contributions on top of that. So together, we get an estimate and can compare it with what we see.

The answer is that it's nowhere close! The estimated value of the vacuum energy is much, much larger than what we observe. So let's think deeply about why that is true. First we need to talk a little about quantum mechanics. This is the correct theory of the world, as far as we know, or at least the correct framework in which to be thinking about the world, and replaces Isaac Newton's classical mechanics.

In both classical and quantum mechanics, you have physical systems that do things. They evolve and obey equations of motion. The real difference between classical mechanics and quantum mechanics is that in classical mechanics, you can observe everything there is to know about the system. So if we have such a system, like a ball rolling down a plane, or a pendulum swinging back and forth, or a set of springs, we can measure where all the components of that system are, and then use the laws of physics to predict where they will be in the future and where they were in the past, in principle to arbitrary accuracy.

In quantum mechanics, meanwhile, there is a rule which says we cannot measure all of the properties that a system has, and in fact you are even dramatically unable to measure them. Think about a coin which you then flip in the air, rotating between heads or tails. While it's still spinning in the air, you can ask whether that coin is still heads or tails. Classically it's somewhere in between, rotating in between either of them. Quantum mechanics is like a coin which you flip and is rotating in the air, yet nevertheless every time you look at it, it's either exactly heads or exactly tails. It's not that we don't know which one it is, but it's that when you're not looking at it, it's some superposition of neither heads nor tails, but is described by some angle. Yet when you look at it, you never see that angle, but you always see it to be exactly head or exactly tails.

For a more realistic example, think about a particle, like an electron. Classically we'd say it has a position. It's located there and has a velocity, moving in some direction. According to quantum mechanics, it's not that we don't know what the position is, but it's that there is no such thing as the position of the electron. What there is, is a function of space, called the wave function which tells us if we look at the electron, where are we most likely to see it? What is the probability of finding it in different places?

Yet the true answer to where the electron is, is not a question that has an answer, since there is no such thing. When you look at the electron, you always see it in a position. Yet when you're not looking, it doesn't have a position. Instead, it has a probability of having different positions. That is the origin of uncertainty in quantum mechanics. What you can observe, is not what there really is. Yet what there really is, is this wave function that tells us a probability of getting different answers to our observations.

So lets apply this to a very specific system, which is not exactly the real world, but is a good analogy to it, and that's a simple pendulum, just a weight that can rock back and forth. It can be stationary, just sitting there, or it can be going back and forth with some amplitude and frequency. This is a good classical kind of thing that Galileo used to look at, when falling asleep in the cathedral in Pisa, where he would time the period with his own pulse, due to lack of a good wristwatch!

Yet these days we've learned to take classical systems and quantize them, to put them in a quantum mechanical framework and see what happens. In the classical pendulum, going back and forth, it has an energy, a potential energy depending on how high its swung up, and a kinetic energy depending on how fast it's moving. Furthermore, those two things, the kinetic and potential energies, can take on absolutely any value. In quantum mechanics, on the other hand, two things happen. One is that the energy cannot take on any value, but just certain discrete values. That's why it's called quantum mechanics. There are specific energy levels you can see, yet you can't see anything in between them.

The other thing is if you ask what the lowest energy level is, you get a different answer in quantum mechanics than classical mechanics. In other words, all you're saying is, here's a pendulum, put it in the zero energy state. Put it in the state of minimum energy. Classically it's kind of obvious what to do, we stop the pendulum from moving. We make it sit there so it's located at the bottom of its trajectory, so its potential energy is minimized and there's no kinetic energy.

Quantum mechanics says that this would be the same as precisely specifying the location of the pendulum, which you simply cannot do. Therefor the minimum energy configuration still has some uncertainty in where it is, and therefor still has some uncertain energy. So this idea of a pendulum is actually quite analogous to the quantum fields we have in the real world. A pendulum classically has energy, yet quantum mechanically it has a slightly higher minimum energy than classically.

In the real world, we have fields that oscillate through space and time. Yet it turns out to be quite a good approximation to think of the field at every point in space as a little pendulum going back and forth. It has a kinetic energy, it has a potential energy, and quantum mechanically it has a little bit more.

Yet we need to notice two things about this analogy. First, if you ask what the energy of the pendulum is, just like the inclined plane example from earlier, there is no right answer. The potential energy of that pendulum would change if we moved the whole apparatus up by a foot, yet none of the physics would change, none of the laws of motion of the movement of the pendulum would be altered in any way.

The quantum fields in spacetime have exactly the same property. There's a minimum amount of energy you can ascribe to them, which is true classically, even in empty space, and it's a completely arbitrary number. Yet then, what quantum mechanics says is that on top of that completely arbitrary number, there's also some quantum mechanical jiggle. Quantum fields have the same property. In addition to the completely arbitrary answer to the question of what the minimum energy is, quantum mechanics adds an extra contribution to that minimum energy, because of the fact that the fields are jiggling back and forth.

This is just Heisenberg's Uncertainty Principle, the fact that you can't localize the quantum mechanical system, without absolute precision. Because of that, there will always be quantum jitters. In the case of a field, like the electron field, the electromagnetic field, or the gluon field, these jitters carry energy and contribute to the ρ of empty space. In other words, they contribute to the vacuum energy.

Now there's a rule, in quantum field theory. When you look at a field, what do you see? Just like for a single wave, you have a wave function, yet when you look at it, you see a particle. For a quantum field, you see a collection of particles. A quantum field is basically a bookkeeping device that tells you how many particles there are, all over the place.

So you might think that if we have a quantum field, its vacuum state, which to physicists means its lowest energy state, the state you can put the quantum field in that has the least energy you can possibly have, would have zero particles. If there are particles, then you have some energy there. The vacuum state of the field, the lowest energy state, should have no particles anywhere and should just describe empty space.

That is correct in the sense that there are no particles in empty space that you can see. Nevertheless, Heisenberg in his Uncertainty Principle, is telling us there are particles that we can't see, which we call virtual particles. The fact you can't pin down the quantum field to some precise configuration, is telling us that you can't avoid virtual particles from popping in and out of existence.

These virtual particles are now a new kind of particle, like we already have electrons and quarks, now we also have virtual particles. They are the good old particles that we know and love, photons, electrons, and positrons, fluctuating in the vacuum, in empty space itself. These virtual particles certainly exist. This is not some crazy, hypothetical thing like it was 70-80 years ago, since we have detected the effects of these virtual particles. They interact with ordinary particles passing through empty space. The fact that these interactions change the atomic energy levels in ordinary atoms, or ordinary stuff, gives rise to corrections in formulas of particle physics, which have been tested to exquisite accuracy.

In other words, we have a picture in quantum field theory of empty space, in which empty space is not boring! It's alive and popping with virtual particles and anti-particles that appear and disappear. We can't see them directly or pull a virtual particle out of the vacuum and make it real, yet we can tell that they're there because they have affects indirectly on the behavior of other particles.

For example, they have affects on gravitons. These virtual particles have a gravitational field, since they carry energy. This energy that they have, is precisely the vacuum energy, a contribution if you like, to the amount of ρ in empty space. Well we said then, that just like for a pendulum, if you ask what the minimum energy of empty space is, there are two contributions. First there is a completely arbitrary, classical number. There's some number there in the laws of physics that says, "If quantum mechanics weren't true, the vacuum energy would be such and such a number." We have no idea what that number is.

We also have a quantum mechanical uncertainty, a zero-point energy of the vacuum fluctuations of every field in the universe, added on top of that. Now what is this energy? How much is the shift in energy from quantum mechanics? There, we can at least estimate it. So even if we didn't know the classical energy, you might imagine that the quantum mechanical shift was of the same basic size as the original vacuum energy.

So it turns out that if you do a naive estimate of how much energy quantum fluctuations adds to the vacuum, you get infinity! So that's not right. Yet in quantum mechanics, such infinities happen all the time, and we know how to fix them. We put on some cutoff, we stop including contributions from very, very short length scales, where spacetime itself may dissolve into some sort of quantum foam.

Then once we have that cutoff , we get a final answer of what the final quantum mechanical contribution to the vacuum energy is. In terms of numbers, it turns out to be 10 to the 112th power of ergs/cm³. That is a lot of ergs (a measurement of energy). If you ask, "Given the data, given the observations from cosmology, how many ergs/cm³ are there?" The answer is 10 to the -8th power of ergs/cm³!

In other words, our best guess, our best estimate on the basis of modern physics, of how much energy there should be in the vacuum, is 10 to the 120th power larger than the amount that we actually see! That's one trillion trillion trillion trillion trillion trillion trillion trillion trillion trillion times bigger than what we actually see! That's a theory that does not match the data, and is called the cosmological constant problem, probably the largest unsolved problem in theoretical physics.

We had known for a long time that there was such a cosmological constant problem, long before we actually detected the vacuum energy. It was obvious that the vacuum energy was not nearly as big as our naive estimate said it should be. So we've had this problem hanging around.

Yet before we detected the acceleration of the universe, the best idea was that there was some secret symmetry of nature, some secret mechanism that took this huge vacuum energy you should have, and exactly canceled it, making it equal to zero. Even though we didn't know what that symmetry was, it didn't seem like such a stretch that someday we'd detect it.

Yet now that we've found some vacuum energy, now that we've found some dark energy that can contribute to the energy density of empty space, the problem has become much harder. We don't want to multiply the real number by zero, but multiply it by 10 to the -120th power!

It's as if you see someone on the street who is flipping a coin, and they ask if we think it will be heads or tails, and we say heads. So they flip it, and it's tails. They flip it again, and it's tails. Then they flip it 299 times in a row, and it's tails every time. You don't know why. You don't know what the mechanism is, but if they then say, "I'm going to flip it again," what do we think it will be? We're going to say tails. Then they flip it, and it's heads!

Something made something go away. Something made the vacuum energy disappear 2 to the 299th times, yet then there's a leftover there, that 300th time. This tiny amount of vacuum energy, by all rights, shouldn't be there, or should be bigger. Yet we're stuck with the situation that it's there.

So it might be right. It might be that once we understand everything, we get a better formula for predicting what the vacuum energy would be, and we get the right answer. Right now, we're simply at a loss. If the cosmological dark energy, is the energy of empty space, it's an idea that fits the data, but about which we have no understanding.

Therefor, we're going to try other things. We still don't know why the vacuum energy is small, but maybe it is zero? Maybe the actual dark energy we are seeing, is something dynamical? That's what we'll be exploring in the next lecture.

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