The Cosmic Landscape

by Leonard Susskind

  Just as the flatbugs living on the two-sphere had no interest in anything but the surface of the sphere, the geometer studying a 3-sphere has no interest in the 4-dimensional space in which it is embedded. We can throw it away and concentrate only on the 3-sphere.

  Einstein’s cosmology involved a space that has the overall shape of a 3-sphere, but like the earth’s surface, the spherical shape is not perfect. In the General Theory of Relativity, the properties of space are not rigidly fixed. Space is more like the deformable surface of a rubber balloon than the surface of a rigid steel ball. Picture the universe as the surface of such a giant, deformable balloon. Flatbugs live on the rubber surface, and the only signals they receive propagate along that surface. They know nothing of the other dimension of space. They have no concept of the interior or exterior of the balloon. But now their space is flexible, and the distance between points can change with time as the rubber stretches.

  On the balloon are markings indicating the galaxies, which more or less cover the balloon uniformly. If the balloon expands, the galaxies move apart. If it shrinks, the galaxies move closer. All of this is fairly easy to understand. The hard part is the jump from two to three dimensions. Einstein’s theory describes a world in which space is flexible and stretchable but has the overall shape of a 3-sphere.

  Now let’s add the element of gravitational attraction. According to both Newton’s and Einstein’s theories of gravity, every object in the universe attracts every other object with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. Unlike electric forces, which are sometimes attractive and sometimes repulsive, gravity is always attractive. The effect of gravitational attraction is to pull the galaxies together and shrink the universe. A similar effect exists on the surface of a real balloon, namely, the tension in the rubber that tries to shrink the balloon. If you want to see the effect of tension, just stick a pin in the rubber.

  Unless some other force counteracts gravitational attraction, the galaxies should start to accelerate toward one another, collapsing the universe like a punctured balloon. But in 1917 the universe was thought to be static—unchanging. Astronomers, like ordinary people, looked at the sky and saw no motion of the distant stars (apart from that due to the earth’s motion). Einstein knew that a static universe was impossible if gravity was universally attractive. A static universe is like a stone, hovering above the surface of the earth, completely motionless. If the stone were thrown vertically upward, then a momentary glance might see it ascending or descending. You might even catch it at the precise instant when it was turning around. What the stone cannot do is to just eternally hover at a fixed height. That is, not unless some other force is acting on the stone opposing the gravitational attraction to the earth. In exactly the same way, a static universe defies the universal law of gravitational attraction.

  What Einstein needed was a modification of his theory that would provide a compensating force. In the case of the balloon, the air pressure from inside is the force that counteracts the tension in the rubber. But the real universe doesn’t have an inside with air in it. There is only the surface. So Einstein reasoned that there must be some kind of repulsive force to counteract the gravitational pull. Could there be a hidden possibility of a repulsive force in the General Theory of Relativity?

  Examining his equations, Einstein discovered an ambiguity. The equations could be modified without destroying their mathematical consistency by adding one more term. The meaning of the additional term was surprising: it represented an addition to the usual laws of gravity—a gravitational force that became increasingly strong with distance. The strength of this new force was proportional to a new constant of nature that Einstein denoted by the Greek letter λ (lambda). Ever since, the new constant has been called the cosmological constant, and it continues to be denoted by λ.

  What had especially caught Einstein’s attention was that if λ were chosen to be a positive number, then the new term corresponded to a universal repulsion that increased in proportion to the distance. Einstein realized he could play off the new repulsive force against the usual gravitational attraction. The galaxies could be kept in equilibrium at a separation that could be controlled by choosing the magnitude of the new constant, λ. The way that it worked was simple. If the galaxies were closely spaced, their attraction would be strong and an equally strong repulsion would be needed to keep them in equilibrium. On the other hand, if the distance between galaxies were so large that they barely felt each other’s gravitational fields, then only a weak repulsion would be needed. Thus Einstein argued that the size of the cosmological constant should be closely connected to the average distance between the galaxies. Although from a mathematical perspective the cosmological constant could be anything, it could be easily determined if one knew the average distance separating galaxies. In fact at that time, Hubble was busy measuring the distance between galaxies. Einstein believed that he had the secret of the universe. It was a world kept in balance by competing attractive and repulsive forces.

  Many things are wrong with this theory. From the theoretical point of view, the universe that Einstein had built was unstable. It was in equilibrium but unstable equilibrium. The difference between stable and unstable equilibrium is not hard to understand. Think of a pendulum. When the pendulum is vertical and the bob is at its low point, the pendulum is in stable equilibrium. This means that if you disturb it a little, for example, by giving it a slight push, it will return to its original position.

  Now imagine turning the pendulum upside down so that the bob is delicately balanced in the straight-up position. If it is disturbed ever so slightly, perhaps by nothing more than the breeze from a butterfly’s wing, the disturbance will build up, and the pendulum will fall over. Moreover, the direction in which it falls is very unpredictable. Einstein’s static universe was like the unstable upside-down pendulum. The slightest perturbation would either cause it to explosively grow or implode it like a popped balloon. I don’t know whether Einstein missed this elementary point or if he just decided to ignore it.

  But the worst thing about the theory was that it was trying to explain something that was just not true. Ironically, there was no need for the new term. Hubble, working with the hundred-inch telescope on Mount Wilson in Southern California, discovered that the universe was not standing still.2 The galaxies were flying apart from one another, and the universe was expanding like an inflating balloon. The forces did not need to cancel, and the cosmological term, which added nothing to the beauty of the equations, could be discarded by setting it to zero.

  But Pandora’s box, once opened, could not be closed so easily.

  The cosmological constant is equivalent to another term that may be easier to picture: the vacuum energy.3 You’ll recall this term from the argument I first encountered at the Belfer School. Vacuum energy sounds like an oxymoron. The vacuum is empty space. By definition it is empty, so how can it have any energy? The answer lies in the weirdness brought to the world by quantum mechanics, the weird uncertainty, the weird granularity, and the weird incessant jitteriness. Even empty space has the “quantum jitters.” Theoretical physicists are used to thinking of the vacuum as being full of particles flickering into and out of existence so quickly that we cannot detect them under normal circumstances. These vacuum fluctuations are like very high-frequency noise that is way beyond the ability of the human ear to detect. But vacuum fluctuations do have an effect on atoms, which like dogs, are much better tuned to the high frequencies. The precise energy levels of the hydrogen atom can be measured to exquisite accuracy, and the results are sensitive to the presence of the fluctuating sea of electrons and positrons in the vacuum.

  These strange violent fluctuations of the vacuum are consequences of quantum field theory and can be visualized using Feynman’s intuitive diagrams. Imagine completely empty space-time with not a single particle initially. Quantum fluctuations can create particles for a short peri
od of time, as in the following figures.

  The first diagram shows an electron and a positron spontaneously created from nothing and then annihilating each other when they come together. You can also think of it as an electron going around a closed loop in space-time, the positron being the electron moving backward in time. The second diagram shows two photons spontaneously created and then annihilated. The last diagram is like the first except that a photon hops between the electron and positron before they disappear. An infinite number of more and more complex “vacuum diagrams” are possible, but these three are more or less representative.

  How long do the electrons and positrons last? About a billion-trillionth of a second. Next imagine these diagrams occurring all over space-time, filling it with a rapidly fluctuating population of elementary particles. These short-lived quantum particles that fill the vacuum are called virtual particles, but their effects can be quite real. In particular, they cause the vacuum to have energy. The vacuum is not the state of zero energy. It is merely a state of minimum energy.

  Back to the Cosmological Constant

  Now a clever reader might ask: “Who cares if the vacuum has energy? If that energy is always present, why don’t we just readjust our definition of energy by subtracting it away?” The reason is that energy gravitates. To understand the meaning of this phrase, you need to remember two easy pieces of physics. The first is (I promised no equations, but I think I’ll be excused for this one) E = mc2. Even schoolchildren know this famous formula that expresses the equivalence between mass and energy. Mass and energy are really the same thing. They are just expressed in different units; to convert from mass to energy, you multiply by the square of the speed of light.

  The second easy piece is Newton’s law of gravity, slightly rephrased in this form: “Mass is the source of the gravitational field.” This is a way of saying that the presence of a mass, like the sun, affects the motion of nearby objects. We can say that either the sun affects the motion of the earth or, in fancier terms, the sun creates a gravitational field, which in turn influences the motions of other objects like the planets.

  Quantitatively, Newton’s law tells us that the magnitude of the sun’s field is proportional to the sun’s mass. If the sun were one hundred times heavier, its field would be one hundred times stronger, and the force on the earth would be one hundred times greater. That’s what it means to say, “Mass is the source of the gravitational field.”

  But if energy and mass are the same thing, then this sentence could also be read: “Energy is the source of the gravitational field.” In other words, all forms of energy affect the gravitational field and, therefore, also influence the motion of nearby masses. The vacuum energy of quantum field theory is no exception. Even empty space will have a gravitational field if the energy density of the vacuum is not zero. Objects will move through empty space as if there were a force on them. The interesting thing is that if the vacuum energy is a positive number, then its effect is a universal repulsion, a kind of antigravity that would tend to drive galaxies apart. This is exactly what we said of the cosmological constant earlier, you will recall.

  This point is so important that I want to stop and explain it again. If, in fact, empty space is filled with vacuum energy (or vacuum mass) it will exert forces on objects that are indistinguishable from the effects of Einstein’s cosmological constant. Einstein’s misbegotten child is nothing but the energy content of the fluctuating quantum vacuum. In deciding to eliminate the cosmological constant from his equations, Einstein was, in effect, claiming that there really is no vacuum energy. But from a modern perspective, we have every reason to believe that the quantum jitters inevitably give rise to energy in empty space.

  If there really is a cosmological constant, or vacuum energy, then there are severe limits on its magnitude. If it were too big, it would lead to detectable distortions of the trajectories of astronomical bodies. The cosmological constant, if not zero, must be very small indeed. The problem is that once we identify the cosmological constant with vacuum energy, nobody has any idea why it should be zero or even small. Evidently, combining the theory of elementary particles with Einstein’s theory of gravity is a very risky thing to do. It seems to lead to an unpromising universe with a cosmological constant many orders of magnitude too big.

  Every kind of elementary particle is present in the violently fluctuating sea of virtual particles called the vacuum. In this sea are electrons, positrons, photons, quarks, neutrinos, gravitons, and many more. The energy of the vacuum is the sum total of the energies of all these virtual particles; each type of particle makes its contribution. Some of the virtual particles are moving slowly and have small energy, while others are moving faster and have higher energy. If we add up all the energy in this sea of particles using the technical mathematics of quantum field theory, we find a disaster. There are so many high-energy virtual particles that the total energy comes out infinite. Infinity is a senseless answer. It’s what made Dirac skeptical of vacuum energy. But as Dirac’s contemporary Wolfgang Pauli quipped, “Just because something is infinite doesn’t mean it’s zero.”

  The problem is that we have overestimated the effects of very energetic virtual particles. In order to make sense of the mathematical expressions, we somehow have to do a better job of accounting for their effects. But we don’t understand much about the behavior of particles when their energy gets above a certain point. Physicists have used giant accelerators to study the properties of very high-energy particles, but every accelerator has a limit. Even theoretical ideas run out of steam at some point. Ultimately we reach a value of the energy so large that if two particles with that much energy collide, they create a black hole! By this point we are far beyond what we can understand with present tools. Even String Theory is not up to the task. So what we do is to make an agreement with one another. We will just ignore the contributions (to the vacuum energy) from all virtual particles whose energy is so large that they would make a black hole if they were to collide. We call it cutting off the divergences or regulating the theory. But whatever words we use, the meaning is the same: let’s just agree to ignore the effects of very high-energy virtual particles that we don’t yet understand.

  It’s a very unsatisfactory situation, but once we do this we can estimate the vacuum energy stored in electrons, photons, gravitons, and all the other known particles. The result is no longer infinite, but it is also not small. The joule is an ordinary unit of energy. It takes about four thousand joules to heat a liter of water 1 degree centigrade. A cubic centimeter is a common unit of volume. It is about as big as the tip of your pinky finger. In the ordinary world the joule per cubic centimeter is a useful unit of energy density. Well then, how many joules of vacuum energy are there in the form of virtual photons in a volume of space as big as the tip of your little finger? The estimate that quantum field theory gives is so big that it requires a 1 with 116 zeros after it: 10 to the 116th power! That many joules of vacuum energy are in your little finger in the form of virtual photons. That is far more energy than it would take to boil all the water in the universe. It’s far more energy than the sun will radiate in a million or a billion years. It is far more energy than all the stars in the observable universe will ever radiate in their entire lives.

  The gravitational repulsion due to that much vacuum energy would be disastrous. It would tear apart not only galaxies but also atoms, nuclei, and even the protons and neutrons that make up the galactic material. The cosmological constant, if it exists at all, must be very much smaller to avoid conflicting with all the things we know about physics and astronomy.

  Now this was just the vacuum energy due to one type of particle, photons. What about virtual electrons, quarks, and all the others? They also fluctuate and create vacuum energy. The precise amount of energy from each type of particle is sensitive to the mass of that particle as well as the various coupling constants. We might expect that if we add the contribution from electrons, it would make the energy eve
n bigger. But that’s not necessarily correct. Photons and other similar particles contribute positive energy to the vacuum. It is one of those paradoxical quantum facts that virtual electrons in the vacuum have negative energy. The photon and electron belong to a class of particles that creates opposite energy in the vacuum.

  These two kinds of particles are bosons and fermions. For our purposes it is not so important to know the detailed difference between these two, but I will take a paragraph or two to explain. Fermions are particles like the electron. If you know any chemistry, you will remember Pauli’s exclusion principle. It says that no two electrons in the atom can occupy exactly the same quantum state. That’s why the periodic table has the structure that it has. As electrons are added to an atom, they fill up ever-higher atomic shells. This is characteristic of all fermion particles. No two fermions of the same type can occupy the same quantum state. They are isolationist hermits.

  Bosons are the opposite, the sociable particles. Photons are bosons. It is especially easy to have many bosons in the same state. In fact a laser beam is an intense collection of photons, all in the same quantum state. You can’t build a laser to make a beam of fermions. On the other hand, you can’t make atoms out of bosons, at least not atoms that have a periodic table.

  What does all this have to do with vacuum energy? The answer is that virtual bosons in the vacuum have positive energy, but virtual fermions like the electron have negative energy. The reasons why are technical, but let’s just accept it as a boon: fermion vacuum energy and boson vacuum energy can cancel because they have opposite signs.