Alan Guth, “Inflationary Cosmology” - LNS46 Symposium: On the Matter of Particles
MODERATOR: I think the statute of limitations has passed on telling stories out of school. So I'm going to do this one. It's 12 years ago or so since when he came. One day, he was at Cornell, had been a student of Francis Low. And one day, he called me up and he said he really wanted to come back to MIT.
We weren't looking for anyone at the time. Did he want to come back to MIT? I hope you'll forgive me, Alan. And he knew that he was too old to come in as an assistant professor. And he'd be willing to take the risk of coming back at a level where the risk of tenure comes sooner.
I brought it up with the theory group, with the high energy group. And we knew that it was a difficult kind of situation. But we did it anyway. We took the risk. Well, here's our risk.
GUTH: Thanks. He collected my first line right away.
Those of you who know me know that I usually have in my title transparency, a cartoon of a well-known cartoon character. But the organizers of this conference took the precaution of having all the speakers sign an affidavit saying that they actually had obtained the right to use all of the visuals that they use during their talks. So sorry, folks. No cartoon. We'll have to do without that.
I'm going to be talking about some of the impact that modern particle physics has had on cosmology. As you probably known, a small drove of particle theorists began to dabble in cosmology back in the late 1970s. And I became part of that drove.
A lot of our colleagues felt we were motivated mainly by jealousy of Carl Sagan. But I've always maintained that there was also a significant motivation, which stems from developments in particle physics itself. And what I have in mind, of course, is the advent of grand unified theories.
These theories were first invented back in 1974. But it wasn't really until the late 1970s that they really began to become objects of widespread interest in the particle theory community. What makes these theories important for cosmology is that they make their most important predictions at an energy scale of about 10 to the 14 or 10 to the 15 GeV. By the standards of our local power company, that's not much energy. It's about what it takes to light a 100-watt light bulb for a minute.
But, as you might imagine, having that much energy on a single elementary particle is, of course, extraordinary. One could imagine trying to build an accelerator that would reach those energies. One could take, for example, the Stanford linear accelerator and imagine scaling it up. The output energy is essentially directly proportional to the length.
So it's a back-of-the-envelope calculation to figure out how long SLAC would have to be in order to reach an energy of 10 to the 14 GeV. The answer, in case you've never done this calculation, turns out to be almost exactly 1 in the right units, the units, of course, being light years. So, needless to say, the Department of Energy and NASA have both been very reluctant to entertain proposals to build such an accelerator.
And what that means is that if we want to be able to see the most dramatic consequences of these grand unified theories, we're really forced to turn to the only laboratory to which we have any access at all, which has ever reached these energies. And that appears to be the universe itself, in its very infancy. And that basically is the reason why so many particle theorists have become interested in the early universe.
Can I have the next transparency? What I want to do in today's talk, since I now have a very interesting experimental result that came out just a few weeks ago, the COBE results, I'd like to basically spend the beginning part of the talk building up to a discussion of COBE. And then I will try to discuss the implications of the COBE measurements.
So I need to begin by discussing the standard hot Big Bang theory. And I know that many of you are nuclear physicists and particle physicists, and have not considered much about the details of cosmology. So I have here a complete summary of the important elements of the standard Big Bang theory in three statements.
The first statement is the assumption that the early universe was a uniformly expanding, extremely hot gas, which was in thermal equilibrium, or at least approximately in thermal equilibrium, at all times during the expansion. So the expansion is just the adiabatic expansion of a thermal equilibrium gas, something which, of course, we've all learned how to treat in our statnet courses.
In the early universe, the mass density, we believe, was completely dominated by black-body radiation, not by the matter that now dominates the mass of the universe. And that makes it, of course, a very simple system to treat. All of the properties of this gas are completely determined once you know the temperature. And you can calculate how fast the temperature changes because you can calculate the dynamics of the expansion and cooling.
The third assumption is what makes that calculation possible. The assumption that the only significant force in the early universe is the force of gravity, which is gradually slowing down the expansion due to the gravitational attraction of everything for everything else. This, of course, in cosmology is generally treated using general relativity. But, in fact, you can use Newton's laws of mechanics. And they work almost as well. And, in many cases, give identical answers.
Next transparency. Okay. So much for standard cosmology. Now, I'll move right along and get into the topic of inflation. Because what I'd like to do is to explain some of the consequences of inflation and to what extent they've been tested by the COBE measurements. So I want to explain to you the mechanism behind inflation because I think it all seems very mysterious, if one doesn't have some notion of how inflation works. So I'd like to go through this reasonably quickly.
Inflation is driven by a scalar field. And the first thing I want to point out is that scalar fields are very commonplace to all particle theorists. They're essential ingredients of the Glasgow-Weinberg-Salam model of electroweak interactions. They're also an essential ingredient of grand unified theories.
The modern superstring theories do not have fundamental scalar particles. In fact, they do not have fundamental particles at all. The fundamental ingredients in those theories are strings. But nonetheless, even in these theories, at energies well below the Planck scale of 10 to the 19 GeV, the theories are believed to be well approximated by our good old-fashioned quantum field theories, in which one has real particles. And in fact, the superstring theories always give rise to a reasonably large number of what appear to be fundamental scalars in this low-energy approximation.
Next transparency. It is typical of a Higgs field that the potential energy function has a minimum at some value other than 0. Now, the field that drives inflation is actually almost certainly not the Higgs field of either the Weinberg-Salam model or the Higgs field of the grand unified theory. But, nonetheless, it is patterned on that model.
So I will assume that the minimum energy of the scalar field is at some non-zero value, here called phi sub t, t for true vacuum. Remember, the definition of the vacuum in this context is simply the state of lowest possible energy density. So that is, by definition then, the vacuum.
However, the potential energy function could also somewhere else have a plateau or perhaps another local minimum. That can be a metastable state. That is, if the plateau is gentle enough, it can take a very long time before a scalar field perched on top of that plateau can find its way to roll down the hill, especially by the time scales of the early universe.
It is that metastable state which is called a false vacuum. That is, a false vacuum is simply a state in some region of space where the scalar field has the value that's here called phi sub f, the value at the top of the plateau. The energy is then dominated by this scalar field. And the energy density I will call rho sub f, f for false vacuum.
Next transparency. I tried to explain here where the word "false vacuum" comes from. It may be more or less obvious by now. A vacuum is being used in the sense of state of lowest possible energy density.
This clearly is not the state of lowest possible energy density. The scalar field could roll down the hill and find the bottom. But the assumption is that that takes a long time. So temporarily, this acts as if the energy is already as low as it can get. So what we have is a temporary vacuum. And the word "false" is being used to denote that notion of temporariness.
Now, just knowing the fact that the energy density of this false vacuum cannot be lowered, at least on an appropriate time scale, that's all one needs to actually derive what is its crucial property for cosmology, which is its pressure. We can derive is pressure by doing a thought experiment. Don't try to actually do this experiment, by the way. It's very expensive, not to mention difficult.
But let's imagine that we have a piston chamber that we filled with a false vacuum, with this non-zero energy density and the pressure which we're trying to. calculate. Outside, we just have ordinary vacuum. So the energy density is 0 and the pressure is 0.
What I want to imagine is that somebody pulls out on the plunger, enlarging the volume of the chamber inside by an amount dV. Now, since the energy density is fixed because the scalar field just has no lower energy state to go to, that it can reach in the appropriate amount of time, the energy density inside this chamber will remain constant, which is obviously something that's very different from any ordinary material that we are used to. If there was a gas inside that, nitrogen, it would just thin out as you increase the volume. But a false vacuum cannot do that. And that's what makes it so different from any other material.
So the energy density remains constant. The energy inside, therefore, goes up by the energy density times the change in volume. By conservation of energy, that must be equal to the work that the person pulling on the plunger has done in pulling it out by that distance. And that work is just minus pdV. And that implies immediately that p has to be equal to minus rho sub f.
In other words, the vacuum creates a suction, a negative pressure, inside this piston. So that to pull it out, you have to do work. And the work you do is just enough so that as you pull it out, the energy density inside remains constant. So what we have is a colossal negative pressure.
Next transparency. Now, what is the cosmological effect of such a negative pressure? Well, one could put this into Einstein's equations to find out what they do to a cosmological model. And what's found is that they act exactly like Einstein's famous cosmological constant, but only temporarily, of course. That is, it only acts like this as long as the false vacuum survives. And it is only metastable, not absolutely stable.
The effect is to slow the cosmic expansion. Well, excuse me. In general, the effect of gravity is to slow the cosmic expansion. And I've written here the key equation that describes how gravity slows the expansion of the universe. The expansion of the universe is described in terms of a scale factor, R. R simply sets the size scale for the entire universe, in the sense that whenever R doubles, it means that the entire universe being described has doubled in size, all distances within it.
So it says that the acceleration of R, R double dot, is negative and proportional to G, Newton's constant, times the mass density plus 3 times the pressure. The first term is purely Newtonian. It's exactly what you would get from Newtonian mechanics.
The second term is what is normally a relativistic correction. In this case, however, it is by no means a relativistic correction. Our pressure is negative and equal in magnitude to the energy density. So the second term dominates and actually changes the overall sign of the entire right-hand side of the equation, thereby completely reversing the effect of gravity.
The false vacuum actually creates a very large gravitational repulsion. And the expansion of the universe is accelerated, if the universe is ever in a false vacuum state, rather than having the expansion of the universe being slowed down, as it normally is, by gravitational effects.
Next transparency. So the key behind the inflationary model is the assumption that the universe went through a short period, during which the matter in the universe was dominated by this very peculiar state that is predicted by many, many of our particle theories, the state called a false vacuum.
To take you through the scenario of the new inflationary universe-- the word "New" here, by the way, as most of you probably know, is there because the original inflationary universe that I proposed in 1981 did not quite work. There was a flaw having to do with the way in which inflation ended. It did not end smoothly enough.
The first working version of inflation was that the so-called new inflationary model, posed independently by Andrei Linde and Albrecht and Steinhardt in 1982. And what I'll basically be describing here today is the new inflationary universe, although now there are several other working versions of inflation, in addition.
The key idea is the belief that a small patch of the universe-- it doesn't have to be the whole universe. And it doesn't have to be very large. 10 to the minus 24 centimeters across will do. A small patch of the early universe somehow settled into a false vacuum state.
There are a number of ideas about how this can happen. In fact, in many of these theories, this is the thermal equilibrium state at high temperatures. So if the universe simply cools quickly, it would supercool into one of these false vacua. In any case, once you assume that this happens for one reason or another, everything else is really pretty well dictated. One doesn't have much in the way of other choices beyond the assumptions that one has to make to get into the false vacuum and start inflation in the first place.
Then this patch of space will start to expand. It expands exponentially. And what you need is for the false vacuum to be sufficiently stable so that that exponential expansion encompasses at least 25 orders of magnitude. It can, by the way, be much more than that. No fine-tuning is necessary to cut that off. No matter how large that is, it's okay as far as the predictions of inflation. But one does need a number at least this large in order for inflation to solve the problems that it was intended to solve.
During this time, the particle density then is essentially driven to 0 by the enormous exponential expansion. If there were any bumps or lumps in the metric of space itself, they are also smoothed out by the enormous expansion. So just like the surface of the Earth always looks flat to us, even though we know it's really round, any fold or bend in the metric of space gets stretched to the point where it becomes completely invisible to any local observation.
There'll be a correlation length in this early universe, a distance over which temperatures, and particle velocities, and values of fields are correlated. That's initially very small, but will be stretched by this enormous expansion. And you need to assume that it's stretched to the length of about 10 centimeters by the time inflation is over. This is the size that what we now call the visible universe would have had at the end of inflation. And that's where this number comes from. Then, the normal evolution of the universe at the end of inflation takes this size scale and stretches it out to be the presently observed universe.
Eventually, the false vacuum decays. It's only metastable, not absolutely stable. When it does, it releases this colossal energy density.
That produces particles, which scatter off of each other, producing more particles. And very quickly, the system reaches thermal equilibrium. And what you have is a hot soup of gas, uniformly spread through space, expanding uniformly, which is exactly what has always been the assumed starting point of the standard cosmological model. So the effect of inflation is simply to establish the initial conditions for standard cosmology, which then takes over at this time.
Next transparency. We need to make the baryons that we see in the universe. Any baryons that might have been present before inflation would get diluted away by the enormous expansion, just like anything else. So it is essential, in an inflationary model, to have an underlying particle physics, in which baryon number is not conserved, so that baryons can be produced after inflation. Fortunately, most modern particle theories have this property. So it does not seem to be any kind of a barrier.
I want to point out that if inflation is right, it means that essentially all of the matter and energy in the observed universe was actually produced during this exponential expansion and then subsequent decay-- I think I lost-- that still works-- then the subsequent decay of the false vacuum. You might wonder what happens to conservation of energy, and I thought you might. So I put it on the transparency here.
What happens to conservation of energy is that it turns out that the gravitational field itself gives a negative contribution to the total energy. And what happens during inflation is that more and more positive energy appears in the form of an enlarged regional false vacuum as the exponential expansion takes place. But at the same time, a compensating amount of negative energy appears in the enlarged volume of the gravitational field. And the net result is that energy is conserved.
It remains constant. And it remains at all times, in fact, incredibly small. And it could even be exactly 0. It is quite consistent with everything we know that the gravitational energy of the universe might precisely cancel the energy that we see in the form of stars, and galaxies, and so on.
Next transparency. You might wonder why people would perhaps believe that inflation may have taken place. Well, inflation does answer questions about a number of the fundamental properties that the universe has, which otherwise go unexplained. Many of these properties are properties that are so obvious that people have grown to accept them without ever feeling that they needed to be explained. But the nice thing about inflation is that it does explain it. And once one has an explanation, one becomes fond of it and one tends to look down on other possibilities in which the explanation gets lost.
So let me just summarize what some of these properties of the universe are that inflation can explain. The first one is simply the bigness of the universe. The universe, as you know, is really very, very, very big. The number of particles, for example, is at least as big as 10 to the 90, which is probably about the largest number you're likely to come across in physics.
The question is simply where did they all come from? They're in the standard cosmological model. Most people probably don't realize this. But you start out assuming that all 10 to the 90 of these particles are already there. The model contains no mechanism whatever for the creation of the particles or of the matter in any form.
In the inflationary model, however, you don't need that. All you need is the small patch of false vacuum, which people now hope might materialize as a quantum fluctuation or something. That's, of course, still very speculative. But in any case, you don't need anything like the 10 to the 90 particles that you need in order to start off a standard cosmological model of the universe.
The second feature is the Hubble expansion, which, of course, is very well known, discovered back in 1929. Inflation leads to it naturally. The exponential expansion is driven by the repulsion of the gravitational effects of the false vacuum and sets the universe out in this motion.
What's going on here is the basic fact that, again, is not very well publicized, that the Big Bang theory does not in any way contain a theory of the bang. The Big Bang theory, as it was developed in the early part of this century, is really the theory of the aftermath of a bang, without any explanation as to what it was that banged. Inflation is an answer to that question. It would be this exponential expansion of the false vacuum, that really would be the bang of the Big Bang.
Third, the question of homogeneity and isotropy. The universe is known to be incredibly uniform. The cosmic background radiation, which is what I'll be coming to shortly, is known to be uniform, same temperature in all directions, to accuracies of one part in 10 to the 5, extreme accuracy.
Now, in the standard cosmological model, that radiation, if you look in different directions, is coming from matter, which at the time the radiation was emitted, those different pieces of matter had not even had time to communicate with each other at the speed of light. In fact, they missed by about a factor of a hundred. So in the standard cosmological model, there is no explanation whatever as to why the temperature should look the same in all directions.
You can make it work in the context of the standard model. You do that by simply assuming that the universe started out with a completely uniform temperature throughout. But what one does not have in the standard model is any explanation whatever as to how or why the universe was so uniform.
In inflation, that happens naturally because one starts with an incredibly small region of space, which has had plenty of time to come to a uniform temperature, the same way a glass of water sitting on the table comes to a uniform temperature. And then inflation takes over, and takes this incredibly small region and magnifies it so that it becomes large enough to encompass the entire observed universe.
Next, the notion of what's called flatness in cosmology. And that is the closeness of the actual density of the universe, to what is called the critical density, which is that density which is just barely sufficient to eventually halt the expansion of the universe. Today, we don't know very well how close we are to the critical density. But even with the big uncertainties in the value today, when you extrapolate it backwards and ask what must it have come from, what you find, if you extrapolate back to, say, one second, which is a very reasonable time in classical cosmology, it's really the beginning of the nucleosynthesis processes, at that time what you compute is that the density must have been equal to the critical density to accuracies of the order of 15 decimal places.
In standard cosmology, one simply has to put that in as an assumption about the initial conditions, without any explanation. In the inflationary model, on the other hand, it turns out that this period of exponential expansion actually drives the universe to a critical density and does it, in fact, to a rather extraordinary accuracy, so that one has a clear prediction that even today the mass density of the universe should be at the critical density if inflation is right.
Finally, there is magnetic monopoles. If grand unified theories are right and standard cosmology were right, it would mean that we would just be swimming in magnetic monopoles. Well, in fact, that we haven't detected or seen any. And inflation gets rid of them by simply inflating them away, diluting them to a negligible density.
Now, I'm ready to move on to the second part of my talk. I want to discuss the cosmic background radiation and what we have learned about it in particular from the COBE satellite. First, I have here simply a side of the spectrum of the cosmic background radiation, as it was measured by COBE.
This is a far better measurement than anything that had preceded it. I had toyed with the idea of showing you some of the earlier graphs just to assure that you'd be adequately impressed by the graph that you're seeing now. I decided that maybe it wasn't worth the time to do that.
But as you can see, this is an absolutely marvelous agreement between the data points, shown as these little boxes, and the black curve, which is a black-body spectrum. Let me remind you that this is only a one-parameter fit. The only thing that's fit is the temperature. Once you know the temperature, you know not only where the peak should be, but exactly how high the peak should be. And both agree perfectly with the data, the same with the entire shape of the curve.
So we have extremely good evidence now that the radiation that we're seeing is remarkably thermal. And that's exactly what would be expected from primordial radiation that is left over from the heat of the Big Bang, which is how this radiation is being interpreted.
Next transparency. Now, I want to just try to describe here exactly where this radiation is really coming from. So what I've drawn is a spacetime diagram of the universe. I'm showing only two dimensions of space, illustrated here as the x- and y-axes, and one direction of time, which is all there is, going upward. The present time is supposed to be this dot. And we can imagine that we are sitting at the position and time of that dot, looking outward.
Now when we look outward, what we see is light that has been traveling from the past. So our past light cone, which is basically the light we're looking at, goes downward in all directions. And the further out we look, as everybody knows, the further into the past we are looking.
Now, let me describe what's supposed to be going on at bottom part of this diagram, which represents the beginning of the time of our universe. The Big Bang is supposed to be at time 0 here. This gray sheet, it's not that easy to see, it's supposed to be a little bit above. It's not at time 0. It's a little bit later. It's supposed to be actually at 300,000 years, is where the gray sheet is supposed to be.
Now, what happened at 300,000 years? The key thing is that according to cosmology, the early universe was incredibly hot. Actually, arbitrarily hot, if you follow it backwards The theoretical t was 0 in the model.
And that means that the matter in the early universe would certainly not have been in the form of neutral atoms. It was a plasma. And until about 300,000 years after the Big Bang, the mass in the universe consisted of a plasma of protons and electrons, and also neutrinos and photons, and so on.
In this hot plasma, photons are constantly being rescattered by the charged particles moving through space. And this plasma is very dense. It's an extremely dense fog, from the point of view of the radiation. And that is what is supposed to exist below this gray sheet.
The gray sheet is supposed to be an equal-time surface, representing the time of recombination, that is the time when matter became neutral. By the way, this has always been called recombination. I don't think anybody really knows why the prefix "re" is there. But this is when protons and electrons combined to form neutral atoms.
Beyond that, suddenly the universe becomes very transparent. Neutral gas is very transparent to radiation. So we believe that the photons that we are looking at now, coming back from along that past light cone, were less scattered where they intersected this sheet at the time of recombination. And this circle, which would really be the surface of a sphere, if I drew a full three-dimensional model, this circle is called the surface of last scattering, the last time this radiation was scattered before the photons set off on their very long journey, which we believe was uninterrupted from then until it is now being received by the COBE satellite, right where my red dot is currently pointing.
Next transparency. Now, what does inflation have to do with these density fluctuations? Well, at first it appeared as if the inflationary model might give a completely uniform universe, with no nonuniformities whatever. And that would, of course, be at odds with our universe.
These nonuniformities, that COBE is detecting, we believe are associated with ripples in the energy density in the early universe, which are in turn associated with the subsequent formation of galaxies, clusters of galaxies, and so on. The key point is that the universe is gravitationally unstable. If there's any slight excess of mass in some region of the universe, that will, of course, create a slightly stronger than average gravitational field, which will in turn pull more mass in, creating a stronger gravitational field. And the process will cascade until one develops the complicated structure of galaxies and clusters that we observe in the universe today. But in order to start it all off, one does need to have small nonuniformities in the initial mass density distribution or else no structure forms.
Now, as I said, the inflationary model, when one first thinks about it, appears to give a completely uniform universe. Because what you have is this energy density of the false vacuum, which is just fixed by the Lagrangian that describes the false vacuum. And it would be the same at all places, giving you a featureless universe. This was a topic that was much discussed back in 1982, shortly after the invention of the new inflationary model by Linde, and Albrecht, and Steinhardt, which brought new interest to the idea of inflation.
It was finally realized-- and I'm pretty sure it was Stephen Hawking who first pointed this out-- that since the world is really quantum mechanical, this classical prediction of inflation is clearly not the final answer. In a quantum mechanical world, any classical statement is, of course, really just the peak of a quantum mechanical probability distribution. So if classical physics, plus inflation, predicts a completely uniform universe, one would assume that quantum physics, plus an inflationary model, would give you small deviations about that uniform mass density. And perhaps these could be the seeds for galaxy formation.
There was then a complicated series of interchanges. Stephen Hawking, himself, wrote a paper about this, claiming that everything worked exactly right. And that you even got the right amplitude for the fluctuations, as well as a spectrum that cosmologists had thought was very plausible.
Then a number of the rest of us started thinking about it. And we didn't understand Stephen's calculation. And we eventually were able to do our own calculations, and got an answer that was very different from Stephen's.
The answer we got was the spectrum was what Stephen had said, which was a desirable result. But that the amplitude that we would get would be about 10 to the 5 times larger than what one actually wanted for galaxy formation. And that was, of course, very disappointing.
This whole issue came to a head in the summer of 1982, when there was a conference in Cambridge, England, which was organized by Gibbons and Hawking, called the Nuffield Workshop on the Early Universe. And about six people or so, they were working on this problem of density fluctuations in the early universe. And we were all disagreeing with each other all during the conference.
But by the end of the conference, we actually did settle our differences and come to an agreement on what the answer is. And that agreement has stuck ever since. And the answer turned out to be not the one that Hawking got, but the other one, the one that gave a number that was 10 to the 5 times too large.
Now, what's also true, though, which is crucial here, is that the amplitude of these fluctuations, it turns out to be very model dependent, very much dependent on the underlying particle physics that one assumes. And we were all working in the context, at this point, of the minimal SU(5) grand unified theory, which at that point was far and away the most popular idea in grand unified theories. And we realized that if that model were not the case, it would be possible to get any amplitude we wanted, including, of course, the right amplitude. So we were spending the next few years hoping that the minimal SU(5) grand unified theory would go away.
Our hopes were fulfilled. Due to the marvelous proton decay experiments, it was found that the prediction that these models made for the proton decay lifetime were inconsistent with observation. And, in fact, now nobody knows what the correct grand unified theory is, if indeed any of them are correct. So what can be said presently is that the spectrum predicted by inflation is about what one wants. And the amplitude is adjustable, depending on what elementary particle physics one puts in to underlie the inflation.
What I now to describe briefly is how one calculates density fluctuations in these models, where they really come from. In an inflationary model, the scale factor driving inflation expands exponentially. The Hubble constant, which turns out to be just R dot over R, is at a constant, chi, the exponential expansion rate. This is to be contrasted with a radiation-dominated universe, in which the scale factor grows like the square root of time. And the Hubble constant then falls, like 1 over 2 t.
Try to remember these equations for the next transparency. Next transparency. This diagram is more or less crucial to understanding just the kinematics of how density perturbations evolve, which is all I'll try to explain today.
First, let's look at the line that my red pointer is now pointing to, which is the inverse Hubble constant. During the inflationary era, that's constant. At the end of inflation, the Hubble constant falls like 1 over t. So the inverse Hubble constant grows like t. So the line goes like this.
Now, the relevance of the inverse Hubble constant is that it sets the effective scale at which different things can influence each other. If two objects in the universe are separated by the inverse Hubble constant, it means that H times the distance is 1. It means that they're moving apart from each other at effectively the speed of light. So things that are beyond H inverse from each other cannot affect each other, while things that are less can. So it sets the scale of causal influence.
The other line follows the wavelength of a typical fluctuation. You follow the Fourier mode as it is stretched by the redshift of the universe. So the wavelength just follows the scale factor itself, which rises exponentially during inflation. And then rises like the square root of time, thereby turning over it and allowing itself to intersect the Hubble constant curve again.
So it took this course twice. Our fluctuation starts out very small compared to H inverse, then becomes larger than H inverse, and then comes back in, inside the Hubble radius again. The notion behind the way fluctuations arise in inflation is that they are established at this point. And then we measure them after they come back inside the Hubble length at much later times.
The fluctuations are caused-- I think I'll skip some transparencies here to save time. But the fluctuations are caused by the nonuniformity with which the scalar field rolls down the hill in the potential energy diagram. It's not a classical process, obviously. It's a quantum process.
And it is possible to really calculate the quantum deviations in the rolling of the scalar field down the hill, which leads effectively to deviations, small deviations, in the time at which inflation ends at different places. And one can trace this through using a linearized approximation of general relativity to actually calculate then what are the density perturbations predicted today.
And then I guess-- why don't you just flash the next few transparencies. I can tell you how many to skip. Audience, don't look. You're not supposed to be bothered with these. Okay, next one.
One more. Hold the next one.
This is the prediction. The prediction is that if you measure a Fourier mode-- so it's a function of k, of the perturbation and the mass density over the mass density, the fractional perturbation of the mass density-- at the time when these waves come back inside the Hubble lens, at the second Hubble crossing in that original diagram, inflation predicts that that should be roughly a constant. It does depend slightly on the underlying particle physics. And if you make the underlying particle physics very strange, you can, in fact, make it more or less anything you want.
But almost all sensible particle physics will give you something here, which is very nearly a constant. And that's known as a Harrison-Zeldovich or a scale-invariant spectrum. And the probability distribution for any given Fourier mode is completely Gaussian. This is actually dictated by just the zero-point quantum fluctuations of a quantum field theory.
Next transparency. Now, we can come to what COBE has measured and how to compare it to this. What COBE has effectively measured are the fluctuations in temperature as one looks at different angles, but with reasonably large errors on any given pixel in this picture. So it's really the data that you get by correlating those pixels that's relevant, and not the map that you get of the pixels themselves.
They've measured it in three different wavelengths, corresponding to 3.3, 5.7, and 9.5 millimeters. For safety's sake, they had two nearly independent channels at each wavelength. So they can compare them with each other. And having three different wavelengths, of course, was also used to check their data.
Let me point out that when you look at black-body radiation, at any given wavelength the intensity gives you a measure of the temperature. It doesn't have to fit the whole curve to figure out what the temperature is. It measures the intensity of any given wavelength. And that uniquely determines the temperature.
So by having three different wavelengths, if all three temperature measurements agree, they think they can have confidence in the results. If they differ, then they might think it's an absorption by some molecule or something. Those kinds of effects would be particular to some frequency or another, and would not give the same temperature shift for all three measurements.
The beam size is quite large. There's a 7 degree Full-Width-Half-Max. So COBE is not capable of seeing the really fine structure in the cosmic background radiation. COBE is only capable of seeing the very, very large-scale structure.
In terms of present-day structures in the universe, this actually is a serious limitation. The clusters and the galaxies that we see actually evolved from perturbations that are too small to be visible to COBE. So what COBE is seeing is the very large wavelength end of the spectrum of perturbations. Perturbations are, in fact, too large in wavelength to be directly responsible for any visible object in the universe.
They were very careful, applied a lot of corrections They even-- this really kind of amazes me. This is on the next-- yeah. They even had to correct for the effect of the Earth's magnetic field on their instruments. After doing all their best shielding, they still found an effect, which they could tabulate and correct for.
And when they were done with all of their subtle corrections, they obtained-- next transparency-- a correlation function, which is the key result which they've obtained. It's the expectation value of the temperature fluctuation at one point, times the temperature fluctuation at another point, at some particular angle, alpha. And this quantity is computed for all pairs of points on the sky and then averaged over all pairs of points at a given value of alpha. And then that's plotted as a function of alpha.
So the correlation function represents the correlations in the fluctuations at a particular angular separation between the two points that are being looked at on the sky. The results are on the next transparency. These are the results. Error bars are shown. The error bars are intended to include both systematic and statistical errors.
If the COBE people are right, once they've refined their data correcting for systematics the way they have, they believe that most of the residual error is, in fact, statistical. But, of course, there's always the possibility of somebody discovering a systematic error that they missed.
The gray band-- and this is what's so beautiful to people who have been working on this-- the gray band represents the prediction of this scale-invariant Harrison-Zeldovich spectrum, predicted by inflation. The reason it is a band and not a line is that the prediction is, after all, probabilistic. What we are seeing here is directly the implications of quantum mechanics.
This is a real quantum mechanical experiment, if our underlying theory is right. So naturally one does not make a precise prediction. But rather, what is predicted by the theory is a probability distribution. And the gray band is the result of Monte-Carlo studies of that probability distribution and represents a 68% confidence level of the distribution.
They fit this to a power law to see how well they could determine this to be a scale-variant spectra. And, unfortunately, the error bars are still pretty large. They fit it to a power spectrum, where n equals 1. The way they've defined things is the scale-invariant spectrum that's being predicted.
And what they find when they do a fit to the data is that they get an n of 1.1, plus or minus 0.5. So it's the right answer. But the error bars are too large. So they have to keep working. And they will.
COBE is still in the air. They have another year's worth of data, which they have yet to analyze, which they certainly will. And they are still in the air, gathering more data. And hopefully, before it's over, they might have as much as four years' worth of data. And the errors should come down considerably.
Next transparency. They also were able to do another test. What they measure is the very long wavelength fluctuations. But we have some idea what the shorter wavelengths fluctuations are from theories of galaxy formation. Those are uncertain by maybe a factor of two. But because they're on such shorter wavelengths, it still gives you a good handle we're trying to extrapolate a slope to determine the variation of the spectrum with wavelength.
So by doing that, and allowing for this uncertainty of about a factor of two in the amplitude at small values of the wavelength, they were able to attain an alternate value of n, which I believe is actually completely independent of the first one. The first one depends on the shape of their own data with wavelength. This just depends on taking the normalization from their data and comparing it with short wavelength estimates from galaxy formation. And what they get this time is an n which is also 1; in fact, to slightly better accuracy, 0.23 this time. And this is a measurement which stretches over three decades of scale.
They also, of course, have measured a magnitude to these fluctuations. They measured the quadrupole moment, which corresponds to a fluctuation in the temperature of 5 parts in a million. This exceeds the best previous bound by about a factor of two, which is very nice. The best previous bound being the result of an MIT balloon experiment by Stephen Meyer. They've also fit the results to an assumed scaling-variant power spectrum, finding a delta T over T, which differs slightly from what they get by measuring the quadrupole.
Next transparency. Now, what are the implications of this? Well-- how much time do I have left? Good.
By being able to directly see these primordial fluctuations in the cosmic background radiation, it enables us to ask a question, which cosmologists have always wanted to answer, but have never before had the ingredients necessary to answer. That is, now that we know how things started, at least on some lens scales, we can ask, do we understand how the universe evolved from these early conditions to the present? In particular, the question tends to be, given these very small nonuniformities in the early universe, can we understand how the very significant structures that we see in the universe evolved?
Well, the answer turns out to be yes under some assumptions, and no under others. And that's very important because it helps single out the theories of structure formation that have a possibility of doing the job. So far, of course, the choice is by no means unique. But a number of possibilities have been ruled out.
In particular, one rather staggering result, but which appears to be valid-- I'm sorry. I have to start the beginning with a caveat. The caveat is important. In order to ask how structure evolved from the primordial fluctuations, one has to look at the fluctuations on the same scale as the structure. And as I mentioned earlier, those are unfortunately not really the scales that COBE is looking at. COBE is looking at much longer wavelengths.
So if one makes no extrapolations, and just tries to work with the raw COBE data as it's measured, one essentially has nothing to say about how a structure in the universe could have evolved because those structures evolved from shorter wavelength primordial fluctuations. So all the implications really hinge on being willing to make the assumption that the scale-invariant spectrum which COBE sees can be extrapolated down to shorter wavelengths. And then, looking at the shorter wavelengths, one can ask, can we understand how those very small deviations in the early universe evolved to become the present-day structure?
So everything I have to say here is based on that one very important assumption. If you make that assumption, then one immediately gets a very remarkable result. All new models in which the mass density is dominated by baryonic matter, matter that's made out of protons and neutrons, appear to be ruled out.
The problem is that baryonic matter couples rather strongly to photons, right up until the time that the matter becomes electrically neutral, this recombination. Then it just turns out there's not enough time for the matter to then clump, if it was that smooth at the time of recombination.
On the other hand, in these models of dark matter, in which the dark matter is assumed to be very weakly interacting, that dark matter would not be strongly interacting with the radiation. And it would start to clump long before the time of recombination, giving itself a head start. So if the dark matter theories are right, the early universe, at the time that COBE was looking at it, was not nearly as smooth as what COBE is seeing. But COBE is seeing a very smooth result because it looks only at the photons. While if the dark matter was not interacting with the photons, it could have been significantly more clumped, and would have been even.
So a strong indication that what we have in the universe is some form of non-baryonic matter. It's still, of course, a big mystery what exactly that matter is likely to turn out to be.
They also looked at a number of specific models. For example, they looked at cold dark matter models, both with large Hubble constants and small Hubble constants. With a large Hubble constant, which is what's discussed here, it does not work. It turns out that there's just not enough time for structure to form if you start out with a-- I'm sorry. I got that wrong. In this model, it actually works the other way. With a large Hubble constant, in a cold dark matter model, too much structure forms, if one assumes the initial conditions as indicated by COBE.
Next transparent-- next and final transparency. Listed here are three models which appear to work. So, as you see, the field is not narrowed down very well yet. What is a very attractive model to most people working in theoretical cosmology is a cold dark matter model with a low Hubble constant, a Hubble constant of 50 in the astronomers' units of kilometers per second for a megaparsec. This is about a low a value as can be tolerated given observations. And some people will say it cannot be tolerated. But some people say that about everything.
The model gives a very good fit to structure on a very wide range of scales. The only thing in structure that does not account for is one observation of very large structure on scales of about a hundred megaparsecs, seen in a study known as the APM study. This has yet to be confirmed. So it's not that clear if a correct model should fit. Many structures have been seen in the universe, and have later gotten away with more careful studies.
There also is perhaps a problem on very short length scales. On very short length scales, the problem is the calculations become very hard to do because things become very non-linear. But people's best estimates give a contradiction on very short length scales.
One possibility for perhaps healing the disagreements of this model would be if one believes, and one does have some reason to believe, that there might be light neutrinos in, say, the seven-electron volt range. Such a neutrino would not dominate the dark matter of the universe. It would not be the dark matter. The dark matter would have to still be some other slowly moving, weakly interacting particle, known generically as cold dark matter.
But a mixture of neutrinos would, of course, give you an extra parameter. And it works exactly in the direction that would allow you to give more power at large scales, and less power at small scales, which seems to be what the astronomical observations are pointing towards. So this model is still probably the favorite of most model builders in cosmology.
But several other models remain open. In particular, number five here, in open model, therefore inconsistent with inflation. But an open model is, nonetheless, apparently consistent with COBE. However, it cannot be an open model in which all the matter is baryonic. It's not the simple universe that some people have advocated. It's still a universe in which more than half of the matter is non-baryonic. It's the only way you can get initially small density fluctuations to grow large enough.
And, finally, what also works quite well, but most particle theorists regard this as distasteful, there is the possibility of having a cosmological constant, which makes up most of the effective mass density of the universe. It's sometimes called vacuum energy. And that does fit the data. Although most theorists regard it as implausible that the cosmological constant should just happen to have a value which is relevant to cosmology, picking up, in particle physics terms, what represents energy of the vacuum, which may not have anything to do with the cosmology.
So as you see, the results of COBE are extremely exciting. And what I think is the most exciting thing is that it's by no means finished. If COBE is right, a number of ground-based, balloon-based experiments are just on the verge of seeing nonuniformities on the cosmic background radiation. And these, in fact, will give us information on the very interesting smaller scales, the scales which really are the scales from which galaxies and clusters of galaxies form.
And the COBE satellite itself, will be also giving us more data. And since most of the errors appear to be statistical, the error bars on the COBE data will come down very significantly over the next years.
So what COBE has done, I think, is to really open a new era in astronomy. We now have a new form of astronomy, the astronomy of the fluctuations of the microwave background. And it has the possibility of telling us a really large amount of information about exactly how our universe evolved. Thank you.