This may be a good time to take a breather as we head down the home stretch and do some accounting. We are in the middle of the third and last large cycle of the course. The first large cycle was on the expanding universe and relativity. The second large cycle was on particle physics, dark matter, and the early universe. The third large cycle is on dark energy and beyond. The smaller cycles of how we got here, why we believe our current theories, and speculations, are sort of different for this final large cycle. We only briefly discussed the supernovae observations and plunged head on into speculations. But it's the nature of dark energy being such a recent discovery for this difference. We don't yet know nearly as much about it as we do the dark matter, so we can't believe in any of our theories and proceed to skip right onto the speculations.
All of our evidence for dark matter and dark energy comes from inferences made by gravity and its effects. Karl Popper famously stated that the best theories are surprising and testable. Relativity is the perfect example, but that does not mean it is somehow too sacred over the years to stop testing. We are confident in its function over small distances of the solar system or the nearest stars, but galactic or cosmological scales are up for scrutiny. Not to mention the fact we all believe it will need modifying on quantum levels of course.
The outer solar system's planets Neptune and Pluto were inferred from gravity before being observed optically One can think of this as being the first dark matter hypothesis. Inner planets were also inferred but turned out to be effects of Einstein's version of gravity. In a similar manner we have now used relativity to infer dark matter on the outer, cosmological scales. But what about the inner, galactic scales of the centers of galaxies?
Modified Newtonian Dynamics (MOND) uses the fact that up to the same point from the center of all galaxies, no dark matter is needed to explain the observed motions. Past that point, gravity can be modifies to account for the observed motions, doing away with dark matter. However on scales of clusters of galaxies like Zwicky examined, there was no way to modify gravity to fit the data. Some form of dark matter was needed to complete the job.
Testing gravity on dark energy is actually easier, since we know so little. We can change the Friedmann equation by setting ρ to zero and having H also start towards zero, but stop before it gets there, accelerating the universe. But the constraints do limit such modifications. The Friedmann equation works very well for the early universe, so any changes will suddenly make it not work so well for the early universe. That tells us that acceleration is a recent phenomenon.
But doing the same thing to Einstein's equation where curvature equals the energy and momentum does not work. Setting ρ to zero as the curvature starts towards zero and then stops, produces all sorts of new particles for gravity besides gravitons.
Even newer tests of gravity use dark energy to predict the structure seen in the CMB and SDSS. Changing relativity to see how it affects the acceleration and evolution of structures of the universe is an exciting project. If dark energy is needed to produce both the acceleration and structure, then Einstein's gravity is confirmed.
The ideas of dark matter and dark energy obviously play an important role in modern cosmology. They've made everything fit together, in the sense that a whole bunch of data of a whole bunch of different kinds of phenomena, suddenly makes sense if you believe that only 5% of the universe is ordinary matter, 25% dark matter, and 70% dark energy.
It's an impressive amount of data, and it's not just the same kind of phenomena over and over again. It's the wide variety of different things we're looking at, that convince us that 95% of the universe is this dark stuff, this dark sector. So physicists are naturally very excited about what this stuff is, what's its properties are. People are buying series of lectures trying to understand where this stuff comes from and what it might be.
However, we have to keep in mind, that every single piece of evidence that we have, points in the direction that there are such things as dark matter and dark energy, come from measuring gravitational fields. We do not find dark matter and dark energy yet, directly in the lab. We hope to do that, and are trying to do that, yet so far it's all been about inferring the existence of dark matter and dark energy, on the basis of the gravitational fields we observe in the universe, on the basis of the way in which space and time are curved, and we attribute that curvature to stuff. We can attribute it to a bunch of cold, massive particles, called dark matter, and a smoothly distributed, nearly constant kind of stuff in empty space, called dark energy.
However, logically speaking, it is certainly possible that our inference is incorrect, because the theory of gravity that we're using to relate observed gravitational fields to the stuff that causes them, is wrong. That theory of gravity of course, is Einstein's Theory of General Relativity. So in this lecture, we'll dig into the possibility that General Relativity is not right. Certainly it's pretty close to being right, both in the solar system, where we've tested it very well, and in other more local things in our galaxy, like a binary pulsar, this pair of two neutron stars which change their rate of rotation due to emitting gravitational waves, just like Einstein predicted.
In other words, we have a lot of empirical data, saying that General Relativity is a pretty good theory on scales of several astronomical units, one light year across or less. Yet now in cosmology, we're looking at the whole universe, many billions of light years across. It's certainly possible that gravity acts differently on length scales of billions of light years, than it does on scales of one light year. Can we do better than Einstein? Can we come up with a better theory than he did, that will explain how gravity works on large scales, and fit the data without invoking dark matter or dark energy?
We should say that Einstein of course is a famous physicist, a very smart guy. Yet even though his theory is very good, there is nothing sacred about it. Therefor there's nothing sacrilegious about questioning General Relativity. There's a feeling that some have of the "establishment cosmologists" having a duty to defend Einstein's honor against attacks from talented but quirky outsiders.
Nothing could be further from the truth. Any working theoretical physicist would like nothing more than to be the one who came up with a better theory of gravity than Einstein! That's why they all got into the game, so we could have a better understanding of nature, something that goes beyond what we already have. Furthermore, we all believe that General Relativity is not the final true answer. We all expect that General Relativity needs to be modified in some way, if only because General Relativity as we currently understand it, is completely inconsistent with our understanding of quantum mechanics.
That's the problem with quantum gravity that string theory has been proposed to resolve, which we'll be getting to in a later lecture. Yet it tells us that General Relativity is at best an approximation. It can't be the final correct answer, because the world is not classical, and General Relativity is. Therefor, maybe General Relativity is also wrong on larger scales?
Yet that's not what we would expect. Usually if we understand something in physics at a different physical distance, we will also understand it at larger distances. New things will kick in at smaller distances where energies become large and quantum mechanics becomes important.
Yet largely speaking, it is a possibility. So we're all very interested in wondering whether or not our current inference about dark matter and dark energy could somehow be traced to a misunderstanding about how gravity works. Before we go into details, it's worth telling this very amusing parable, this story about how people have already tried this game, and how we get ambiguous results from looking at the historical record.
Most of what we know about gravity, both even today, as well as in the past, comes from the dynamics of stuff in the solar system. The different satellites around the planets, and the planets moving around the sun, all moving in a gravitational field. This is how Kepler was able to find out that the planets move in ellipses rather than in perfect circles, and from planetary motions, Isaac Newton was able to derive his inverse square law for gravity, t he idea of the gravitational force between two objects, decreasing inversely as the distance squared.
However, this theory of Newton's, even though it fit the data of planets very well, didn't always fit perfectly. The first time anyone discovered a new planet since ancient times, was that of Uranus in 1781. It was discovered with a telescope, a brand new planet out there, once again showing evidence the heavens were not fixed and given to us ahead of time. We had to go out and look to see what would be there.
So people were very excited about this, and took many different observations of the motions of this new planet, and tried to ask whether or not it fit into what we understood about the solar system. Interestingly, these motions were not precisely those expected from the laws of gravity as handed down by Newton. So what do you do when there is an object in the sky, whose dynamics are not exactly what you would have predicted, on the basis of the gravitational fields you think are there?
You invent dark matter! You invent some new stuff you haven't directly seen, whose gravitational field is influencing the motion of the things you do see. So the French mathematician Urban Le Verrier, invented a new planet that we eventually called Neptune. He predicted exactly where it would have to be located in the sky, in order for it to be pushing around the orbit of Uranus in the right way. When he finally convinced people to go look for it, they found it to within one degree from where he predicted it to be on the sky.
This was the first example in history of where a dark matter hypothesis was presented and turned out to be right. Of course when you see it, it's no longer dark matter, but we're hoping to do the same exact thing with real dark matter and real dark energy.
Then it was realized within Le Verrier's lifetime, that there was also a problem with the inner planets. The motion of Mercury in particular was not precisely what Newton said it should be. It moves in an elliptical orbit that was known to precess, to change its orientation gradually as a function of time. Part of this precession is due to the solar system not being a pristine place. There's more to it, then just the Sun and Mercury. The sun is rotating itself, which gives it a different kind of gravitational field. Also of course, there are the other planets.
So Le Verrier, fresh off of his success of predicting dark matter and discovering Neptune, did the same trick again. He proposed there was dark matter in the inner solar system, in the version of a new planet called Vulcan. So it would be another minor planet in the solar system, inside the orbit of Mercury. Le Verrier predicted exactly where it should be, and lo and behold it was discovered, in fact more than once. All these times it was discovered turned out not to be right, since of course there is no planet there.
In the case of Mercury, the reason why it's not moving in the way Newton predicted, was because he wasn't right! Gravity doesn't quite work the way he said it did. There are slight deviations which are understood when Einstein came along and made his General Theory of Relativity. After that, one of the first things he did, was to check what it would predict for the orbit of Mercury and he found that it exactly counted for the discrepancy from what was observed and predicted. Einstein said this was the happiest moment of his life, when he realized before anyone else in the world, the correct explanation for the orbit of Mercury.
So what is the lesson for our little story? We can see that things move through the heavens, and then use them to predict the existence of stuff, based on our understanding of gravity. Sometimes that prediction turns out to be right, so that we do understand what gravity is doing, and we successfully infer the existence of new stuff. Yet sometimes it's because gravity is the new culprit. So we can try doing the exact same thing now, and try to use a new theory of gravity to try and explain away dark matter and dark energy.
As one final bit of caution, we should notice that in the solar system, one place that the dark matter hypothesis worked, was in the outer solar system. The place where we didn't understand gravity was at shorter distances, smaller length scales. So the fact we understand gravity right now, very very well in the solar system, suggests we probably understand it in the galaxy and in cosmology. Once again because the very discovery of dark matter and energy have been such a surprise, we're trying to keep and open mind to see what will happen.
So the evidence we have for dark matter and dark energy, takes two very different forms for those two different things. We believe that dark matter is something local, that falls into places where there is ordinary matter, into galaxies and clusters, and increases the strength of the gravitational field. It's a fairly simple idea to try to get rid of dark matter with a modified theory of gravity. The fact that you have a galaxy or cluster and you measure its mass by looking at the motions of things near the edge, and you find too much, could potentially be explained by the simple hypothesis that the gravity is stronger at large distances than you thought it to be. In other words, it falls off more slowly than 1/r² that Newton would have us think. That's the kind of thing you'd need to make work, if you want to replace dark matter with gravity.
Dark energy on the other hand is smooth and global. It does not collect into galaxies and clusters, but is all over the place. Furthermore it does not pull things together more strongly, like dark matter does. It is pushing the universe apart. The first manifestation of energy was that the universe is accelerating at a faster and faster rate. So to get rid of dark energy, rather than dark matter, we need to find something that makes the universe accelerate. Something that's also not associated with any kind of source, so it's all over the place and not just collected in the area where there is ordinary matter.
So the attempts to do away with dark matter and dark energy by modifying gravity, seem to have a different character. Maybe they're the same modification or maybe they're two different modifications. Maybe there is dark matter but not dark energy, or vice versa, all these possibilities are there on the table.
So lets think in more detail about the dark matter case. What do we actually know? What is our evidence for dark matter? The original evidence of course, came from clusters and galaxies. We look at the dynamics of these systems, stuff moving around galaxies for example. We use that stuff to infer the total amount of mass. We look at the motions of galaxies, use the velocities at which they are moving to infer the total amount of mass in a cluster of galaxies.
These days we have slightly more sophisticated forms of evidence that are slightly less direct. We have the CMB, the different splotches of which respond to the existence of dark matter. That's more evidence that there's more in the universe than just ordinary matter. We have the evolution of structure in the universe, and for that matter, we have the evidence that the universe is structurally flat, which relies to some extent on the existence of dark matter.
Now it's kind of remarkable that the one hypothesis of dark matter particles, fit all those different phenomena. It would be very difficult to imagine coming up with a new theory of gravity that did just as well. So we can start with a simple starting point. Forgetting about trying to explain everything at once, let's just take one phenomenon and try to explain that.
So this was done in 1984, by Mordehai Milgrom who said, "Lets just look at the rotation curves of spiral galaxies." Remember those were measured by Vera Rubin in the 1970s, giving us the first really quantitative evidence that most of the stuff in the universe was dark matter and not ordinary matter. Rubin measured the velocity of different kinds of stuff as you went from the center of a galaxy to the far edges. If what you saw in the galaxy was what there was, that velocity would fall off as you got further and further away. Rubin found that this didn't happen, and the velocity stayed nearly constant. This was evidence that you needed dark matter in the outskirts of the galaxy, to make the rotation curves make sense in the context of Newtonian gravity.
Milgrom noticed a very interesting fact about the universe. Regardless of whether or not he's right about gravity, this fact remains true. If you look at different galaxies and measure their rotation curves, then look at where you need dark matter and where you don't, you find that in the centers of galaxies you don't need to imagine there's dark matter. Maybe it's there, but the ordinary matter is enough to explain the motions you observe. It's only in the outskirts that the effect of dark matter builds up enough, that you absolutely imagine that it's there, to explain the data.
Now the radius of the region where ordinary matter is enough, is different for every galaxy. Everyone knew that fact already, yet Milgrom noticed that if you calculate, just using Newton's laws, the acceleration due to gravity at the point where ordinary matter is no longer sufficient, the point where you need to invoke dark matter to explain the motions, that acceleration due to gravity was the same in every galaxy. There was some crossover value of the acceleration such that when the acceleration of gravity was larger than that value, you didn't need dark matter. When it was smaller, you did need dark matter.
So Milgrom thought perhaps there wasn't any dark matter, but a new theory of gravity instead? Maybe the theory of gravity that was correct, had the property that if the acceleration is greater than a certain value, it's a 1/r² law, just like Newton said. Yet at smaller values of the acceleration, gravity has a force that is stronger than that, so if falls off more gradually like 1/r rather than 1/r².
Well whether or not that's true, it fits the data very well, not only for the galaxies that Milgrom knew about when he was making this hypothesis, but also for new kinds of galaxies that were discovered after he made it. For individual galaxies, Milgrom's rule of thumb, that you don't need dark matter and Newtonian gravity works in the insides with a very specific radius outside of which you can fit the data by modifying gravity, fits over and over again. This is a remarkable fact about the universe, maybe because there's modified gravity, maybe because cold dark matter arranges itself so that will be the case. If the latter possibility is right, it's a challenge for the theory of cold dark matter to predict why this observed fact found by Milgrom, seems to be so true in the real world.
The problem with Milgrom's theory, called MOND (Modified Newtonian Dynamics) is that it's not a theory. It's a suggestion for one very specific circumstance, for the rotation curves of galaxies. Yet we know there's more to the universe than galaxies. There are clusters of galaxies, there is the expansion of the whole universe. There are more phenomena than rotation. There are galactic gravitational lenses for example. How can you make predictions for all of these observable phenomena, from Milgrom's theory?
The answer is you can't. You need to take that suggestion that the law of falloff due to the gravitational force, is different than Newton's, and embed it in a more comprehensive framework, from which you can make better predictions. Even before you do that, the one thing you can do is try to go from galaxies to clusters of galaxies. What we've found over and over is that it doesn't work. Milgrom's modified rule works very well for individual galaxies, yet when you start including clusters of galaxies, you could make a prediction, and it did not come true. People tried and tried to understand clusters of galaxies within Milgrom's framework, and they never quite succeeded.
Eventually they more or less gave up on the idea that you could explain clusters of galaxies purely by modifying gravity, at least with Milgrom's idea. The people who are now proponents of MOND, believe that in order to explain clusters of galaxies, there has to be some dark matter there. However, they say that the dark matter within clusters, might be different than the kinds of dark matter you imagine in conventional concordance cosmology.
For example, the dark matter in clusters of galaxies, could be neutrinos. In ordinary cosmology, we don't think the dark matter could be neutrinos because they are hot dark matter. They move very quickly, so don't settle into galaxies and make them form. In a universe dominated by neutrinos, structure would be very featureless. Yet they do move slow enough that they could be captured by clusters of galaxies. So basically, people who believe in Milgrom's theory have retreated to a position where clusters of galaxies are explained by dark matter, but individual galaxies are explained without dark matter, by a modified theory of gravity.
Still, it wasn't until 2004, 20 years after Milgrom's original theory, that Jacob Bekenstein finally managed to come up with a full theory that reduced to Milgrom's idea under the appropriate circumstances. We'll not be surprised to learn that this theory was very complicated. It's not easy to come up with a modification of gravity that fits all the data. So Bekenstein's theory invented new fields. That's what you do whenever you invent a new theory of anything, you add new fields and see that they do. He added a bunch of new fields, found that he could predict what Milgrom predicted for the cases of individual galaxies, yet could also fit other cosmological data.
Remarkably, Bekenstein's theory fit things like the CMB and the acceleration of the universe, if you not only put in the neutrinos you needed to explain clusters, but also dark energy. So on the one hand, you finally had a model of modified gravity in which individual galaxies did not have important amounts of dark matter in them, yet other than that, it was a very conventional model.
Clusters of galaxies and the whole universe are defined by dark matter and dark energy. So even though that kind of model fits the data, the people who were originally enthusiastic about it have lost a certain amount of motivation. Originally the hope was that you were getting rid of dark matter entirely. Now you're just getting rid of non-standard model dark matter, hoping that it can be neutrinos. The motivation for doing that is a little bit less. Yet it still could be true.
Let's back up a little bit from the notion of Milgrom's particular proposal, to the general notion of doing away with dark matter in galaxies. Sean mentions his believe here, that you absolutely need to have some kind of dark matter in the universe. It's impossible in principle, to think of a theory in this day and age, which will completely do away with dark matter. There are two pieces of evidence for that.
One is the Bullet cluster of course, where we mentioned the original evidence for dark matter. Here you have two clusters of galaxies that pass through each other. The hot gas that is most of the ordinary matter, got stuck in between, while the galaxies and dark matter went right on through. When you look for the gravitational fields of these clusters, which you find with gravitational lensing, you then see that most of the gravity is being caused by the dark matter, not by the ordinary matter that got stuck in the middle.
If you're going to modify gravity in a way to get rid of dark matter, typically you will modify the strength of gravity as a function of distance. Yet it's very hard to modify the direction of gravity. If you're trying to come up with a model in which there is no dark matter, the only source for gravity is the ordinary matter. The only plausible direction for which the gravitational force can point in, is toward the ordinary matter. Yet the Bullet cluster is a very clear-cut example, where we have a gravitational field that is pointing in a direction other than where the ordinary matter is. So the Bullet cluster makes perfect sense if there is dark matter, yet is very hard to understand if there is not dark matter and gravity is modified.
Similarly you have the CMB. We told the story of the oscillations of over-dense regions in the early universe, and how sometimes they were in phase with the dark matter, while sometimes they were out of phase. That is precisely what you observe in the CMB. If there is no dark matter, if there is just modified gravity, then there isn't anything to be in phase with, or out of phase with. It becomes correspondingly much harder to explain what we see in the CMB.
These are both very general arguments. They don't attack any specific proposal, and what they say is that they have very good evidence that there isn't any specific proposal that you can possibly come up with, to do away with dark matter and replace it with modified gravity. You never know without a theorem that is absolutely rigorous, but is pretty good evidence that dark matter really does exist.
So what about dark energy? This is a very different case from dark matter. Among other things, it's much simpler, or at least we know much less about it. Dark matter clusters together and has dynamical properties, while dark energy only might be dynamical. The only thing that we know is that it's a nearly constant ρ, yet enough to make up 70% of the critical density of the universe.
So what you want to do to modify gravity in order to get rid of dark energy. In other words, imagine dark energy is zero, but there's some modification of Einstein's equation that makes the universe accelerate. Ultimately you modify the Freidmann equation. That's the one that tells how the expansion rate of the universe (H), responds to energy density (ρ).
ρ = H²
It tells us that if ρ goes to zero, then H goes to zero along with it. So if you want to do away with dark energy, you want to modify the Friedmann equation in such a way, that as ρ goes to zero, H stops going to zero. Either it goes to zero much more slowly than ρ does, or it actually gets stuck at some finite value.
The important thing to realize here is that there are constraints on what you can do. Just like with dark matter, there are plenty of experimental constraints, so for dark energy we're going to modify the Friedmann equation. There are some things we know about the Friedmann equation, for example we know it works very well in the early universe. It's been tested to very good precision by both primordial nucleosynthesis and the CMB. We have two sets of phenomena, which if you try to explain them on the basis of the Friedmann equation plus ordinary matter and radiation, you get exactly the right answer.
If you try to modify the Friedmann equation, for example, try to imagine modifying it so that the universe is always accelerating. Then you would be in dramatic disagreement with predictions from Big Bang nucleosynthesis and the CMB. All of the data that we have today, are telling us that the acceleration of the universe is a recent phenomenon, cosmologically speaking. When the universe was half it's current size, it was the time when it went from decelerating due to the matter and radiation in it, to accelerating due to the dark energy.
So if you want to modify the Friedmann equation, you do so in such a way that it "turns on" relatively late in the universe's history. Yet here's the problem with that idea. It can be done, but there are issues we need to face up to. Here we see Einstein's equation again, telling us that the curvature of spacetime on the left, is related to the stuff of the universe on the right:
Rμν - ½Rgμν = -(8πG/c^4)Tμν
You want to modify this equation such that when the stuff goes away, there is still spacetime curvature. Even if there's no energy density, even if the universe expands and dilutes away, all the matter and radiation, spacetime curvature will not go to zero, that is you goal. The problem is that this equation of Einstein's is actually quite remarkable. If you try to mess with it just a little bit, you basically just break it.
What we mean by this, is if you try to add terms to Einstein's equation that are small, they give rise to some discrete, new effects. In particular they turn on new fields. We mentioned very briefly that the force due to gravity in a quantum mechanical context, can be thought of as due to the exchange of gravitons, spin 2, massless bosons. Yet if you change Einstein's equations in any relevant way, there are more particles than just gravitons. There are new kinds of particles that qualify as part of the gravitational force. Those particles have an effect.
So you're trying to mess with Einstein's equation just a little bit, just so it's important in cosmology at late times when the universe wants to begin to accelerate, yet is completely irrelevant here in the solar system. What you find is that's much harder to do than you thought, at least the simple thing is that you do have a very interesting effect. We can test General Relativity in the solar system, and ask whether or not the fields that are there, that are affecting the spacetime metric, are the ones that we're observing.
So we see a picture of the Cassini probe, a satellite launched by NASA which nominally had as its job, to take pictures of Saturn, Jupiter, and the other outer planets. Yet along the way, it did us a favor by beaming signals from itself to us here on earth, as it was passing through the path of the sun. So we had the sun, and behind the sun was this spacecraft. it passed signals to us, and one of the predictions of General Relativity is that time is warped along with space. As a result of this, the time it takes the signal to get to us, is delayed by the gravitational field of the sun.
We know the exact trajectory of the satellite, due to the high-precision planning for getting it to Saturn correctly. So we can do a very good job of measuring that time delay, which turns out to be precisely what General Relativity predicted. This means there aren't any new gravitational fields in the solar system, of the kind you'd predict if you try to mess with Einstein's equations.
For whatever reason, Einstein's equation is very true in the solar system. Yet the rule that we have, is that if you want to change something on large scales, it's very hard not to change them on small scales. No one has been able to come up with a nice theory where we are able to change Einstein's equation to make the universe accelerate at very late times, without modifying the dynamics right here in the solar system. You might expect it to be possible, but we haven't been able to do it yet.
On the other hand, to take these lemons and make lemonade, by thinking about these theories, we've been inspired to new ways to test General Relativity. What we have in the case of dark energy, is an accelerating universe, and from the dark energy hypothesis, and the amount of acceleration we observe, we can make predictions for how structure should evolve as the universe gets bigger. We have an early universe which is very smooth, with small perturbations in ρ from place to place, that you can see in the fluctuations of the CMB, and under the force of gravity those fluctuations grow into large-scale structure today.
We see a map of the SDSS that the distribution of galaxies across the sky, which come from the tiny ripples we see in the CMB. The equations that relate the early, tiny ripples, to what we see today, are of course Einstein's equations. So the point is, if you mess with Einstein's equation in order to make the universe accelerate, you will also be changing the equations that make structure grow.
So if you try to calculate the effects of dark energy on both the acceleration of the universe and on the growth of structure, and then you go out and observe those two things separately, and find they are incompatible, it is possible to explain that incompatibility by imagining that gravity has been modified. Or turning around the same logic, if you find that the same dark energy theory explains both the acceleration of the universe, and the evolution of structure, then Einstein was right.
So right now, we're trying to do this. It's one of the things that cosmologists are embarked on as we speak, to try and measure not only the expansion of the universe using supernovae as standard candles, but also the evolution of structure in the universe. If you get a consistent picture by imagining that there is Einstein's equation, dark energy, and dark matter, we'll know that Einstein, who was very smart after all, will probably be having the last laugh.