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The End of the World
Some say the world will end in fire,
Some say in ice.
Some say in ice.
—ROBERT FROST, “Fire and Ice”
M
y account of the state of the universe would be incomplete without a description of how the world is going to end. The theory of inflation tells us that the universe as a whole will go on forever, but our local region—the observable universe—may well come to an end. This issue was at the center of cosmological research for a good part of the last century, and our picture of the end has changed several times in the process. I will now review the recent history of the subject and then give you the latest on cosmic eschatology.
y account of the state of the universe would be incomplete without a description of how the world is going to end. The theory of inflation tells us that the universe as a whole will go on forever, but our local region—the observable universe—may well come to an end. This issue was at the center of cosmological research for a good part of the last century, and our picture of the end has changed several times in the process. I will now review the recent history of the subject and then give you the latest on cosmic eschatology.
After Einstein denounced the cosmological constant early in the 1930s, the predictions of Friedmann’s homogeneous and isotropic models were clear and simple: the universe will collapse to a big crunch if its density is greater than critical and will continue expanding forever otherwise. To determine the fate of the universe, all we had to do was to accurately measure the average density of matter and see whether it is greater than critical. If it is,
then the expansion of the universe will gradually slow down and will be followed by contraction. Slow at first, the contraction will accelerate. Galaxies will get closer and closer, until they merge into a huge conglomerate of stars. The sky will get brighter, not because of the stars—they will most probably all be dead by then—but because of the increased intensity of the cosmic background radiation. The radiation will heat up the remnants of stars and planets to very uncomfortable temperatures, and any living creatures who managed to survive until then will end their days like lobsters in boiling water.
then the expansion of the universe will gradually slow down and will be followed by contraction. Slow at first, the contraction will accelerate. Galaxies will get closer and closer, until they merge into a huge conglomerate of stars. The sky will get brighter, not because of the stars—they will most probably all be dead by then—but because of the increased intensity of the cosmic background radiation. The radiation will heat up the remnants of stars and planets to very uncomfortable temperatures, and any living creatures who managed to survive until then will end their days like lobsters in boiling water.
The stars will eventually be disrupted in collisions with one another or get vaporized by the intense heat of the radiation. The resulting hot fireball will be similar to the one in the early universe, except now it will be contracting rather than expanding. Another difference from the big bang is that the contracting fireball is rather inhomogeneous. Denser regions collapse first to form black holes, which then merge into larger black holes, until they all merge together at the big crunch.
In the opposite case of less-than-critical density, the gravitational pull of matter is too weak to turn the expansion around. The universe will then expand forever. In less than a trillion years all stars will exhaust their nuclear fuel. Galaxies will turn into swarms of cold stellar remnants—white dwarfs, neutron stars, and black holes. The universe will be completely dark, with ghostly galaxies flying apart into the expanding void.
This state of affairs endures for at least 10
31
years, but eventually nucleons that make up the stellar remnants decay, turning into lighter particles—positrons, electrons, and neutrinos. Electrons and positrons annihilate into photons, and the dead stars begin slowly to dissolve. Even black holes do not last forever. Hawking’s famous insight, that a black hole leaks out quanta of radiation, implies that it gradually loses all its mass, or, as physicists say, “evaporates.” One way or another, in less than a google years all familiar structures in the universe will be gone. Stars, galaxies, and clusters will disappear without a trace, leaving behind an ever-thinning mix of neutrinos and radiation.
1
31
years, but eventually nucleons that make up the stellar remnants decay, turning into lighter particles—positrons, electrons, and neutrinos. Electrons and positrons annihilate into photons, and the dead stars begin slowly to dissolve. Even black holes do not last forever. Hawking’s famous insight, that a black hole leaks out quanta of radiation, implies that it gradually loses all its mass, or, as physicists say, “evaporates.” One way or another, in less than a google years all familiar structures in the universe will be gone. Stars, galaxies, and clusters will disappear without a trace, leaving behind an ever-thinning mix of neutrinos and radiation.
1
The fate of the universe is encoded in the parameter called Omega, defined as the average density of the universe divided by the critical density. If Omega is greater than 1, the universe will end in fire and a big crunch; if it is less than 1, we can look forward to freezing and slow disintegration. In the
borderline case of Omega equal to 1, the expansion gets increasingly slow, but never stops completely. The universe narrowly escapes the big crunch, only to become a frozen graveyard.
borderline case of Omega equal to 1, the expansion gets increasingly slow, but never stops completely. The universe narrowly escapes the big crunch, only to become a frozen graveyard.
For more than half a century astronomers worked very hard trying to measure the value of Omega. However, nature was not in the mood to disclose her long-term plans. Omega was tantalizingly close to 1, but the accuracy of measurements was not sufficient to tell whether it was above or below.
Our view of the end was transformed in the 1980s, when the idea of inflation appeared on the scene. Prior to that, a big crunch and an unlimited expansion seemed equally probable, but now the theory of inflation made a very definite prediction.
During inflation the density of the universe is driven extremely close to the critical density. Depending on quantum fluctuations of the scalar field, some regions have density above and other regions below critical, but on average the density is almost exactly critical. Thus, anyone who had nightmares that the universe might collapse to a big crunch in a few trillion years could now relax. The end will be slow and unexciting, as the cold remnant of our Sun hangs around for eons, waiting for all of its nucleons to decay.
A characteristic feature of the critical-density universe is that the structure formation process is stretched over an enormous length of time, with larger structures taking a longer time to assemble. Galaxies form first, they later clump into clusters, and still later the clusters clump into superclusters. If the average density in our observable region is above critical, then in about a hundred trillion years this entire region will turn into a huge super-duper cluster. By that time all stars will already be dead and all observers will probably be extinct, but structure formation will continue, extending to larger and larger scales. It will only stop when the cosmic structures disintegrate as a result of nucleon decay and black hole evaporation.
Another twist introduced by inflation is that the entire universe will never come to an end. Inflation is eternal. Countless regions similar to ours will be formed in other parts of the inflating spacetime, and their inhabitants will struggle to understand how it all began and how it will end.
Friedmann’s relation between the density of the universe and its ultimate fate applies only if the vacuum energy density (the cosmological constant) is equal to zero. This was the standard assumption before 1998, but when evidence to the contrary was discovered, all earlier forecasts for the future of the universe had to be revised. The main prediction that the world will end (locally) in ice rather than in fire did not change, but some details had to be modified.
As we discussed earlier, the expansion of the universe begins to accelerate once the density of matter drops below that of the vacuum. Any gravitational clumping stops at that time. Clusters of galaxies that are already bound together by gravity survive, but looser groups are dispersed by the repulsive gravity of the vacuum.
Our Milky Way is bound to the
Local Group
, which includes the giant spiral of Andromeda and some twenty dwarf galaxies. Andromeda is on a collision course with the Milky Way; they are going to merge in about 100 billion years. Galaxies beyond the Local Group will all speed away, moving faster and faster. One by one, they will cross our horizon and disappear from view. This process will be complete a few hundred billion years from now. In that remote epoch, astronomy will become a very boring subject. Apart from the giant galaxy resulting from our union with Andromeda and its dwarf satellites, the sky will be completely empty.
2
We should enjoy the show while it lasts!
Local Group
, which includes the giant spiral of Andromeda and some twenty dwarf galaxies. Andromeda is on a collision course with the Milky Way; they are going to merge in about 100 billion years. Galaxies beyond the Local Group will all speed away, moving faster and faster. One by one, they will cross our horizon and disappear from view. This process will be complete a few hundred billion years from now. In that remote epoch, astronomy will become a very boring subject. Apart from the giant galaxy resulting from our union with Andromeda and its dwarf satellites, the sky will be completely empty.
2
We should enjoy the show while it lasts!
Our forecast for the universe would now be complete if the cosmological constant were truly a constant. But as we know, there are good reasons to believe that the vacuum energy density varies in a very wide range, taking different values in different parts of the universe. In some regions it is large and positive, in other regions large and negative, but only in rare regions where it is close to zero will there be some creatures who care about what it is.
It follows that the value we observe here is not the lowest possible energy
density; inevitably, it will get smaller in the future. Consider, for example, Linde’s model, where the vacuum energy is due to a scalar field with a very gently sloping energy landscape (
Figure 13.2
). The slope is so small that the field changed very little in the 14 billion years since the big bang. But eventually it will start rolling downhill, and cosmic acceleration will begin slowing down. At some point the field will get below the zero level, to the negative values of energy density. A negative-energy vacuum is gravitationally attractive, so before long cosmic expansion will stop and contraction will begin.
density; inevitably, it will get smaller in the future. Consider, for example, Linde’s model, where the vacuum energy is due to a scalar field with a very gently sloping energy landscape (
Figure 13.2
). The slope is so small that the field changed very little in the 14 billion years since the big bang. But eventually it will start rolling downhill, and cosmic acceleration will begin slowing down. At some point the field will get below the zero level, to the negative values of energy density. A negative-energy vacuum is gravitationally attractive, so before long cosmic expansion will stop and contraction will begin.
An alternative scenario, suggested by the string theory landscape picture, is that classically our vacuum is stable and has a constant energy density, but quantum-mechanically it can decay through bubble nucleation. Bubbles of negative-energy vacuum will occasionally pop out and expand at a speed quickly approaching the speed of light. A bubble wall may be charging toward us at this very moment. We are not going to see it coming: the light will not get much ahead of the wall, because it is moving so fast. Once the wall hits, our world will be completely annihilated. The particles that make up stars, planets, and our bodies will not even exist in the new vacuum. All the familiar objects will be instantly destroyed and turned into clumps of some alien forms of matter.
One way or the other, the vacuum energy will eventually turn negative in our local region. The region will then start contracting and will collapse to a big crunch.
3
Exactly when this is going to happen is hard to predict. The rate of bubble nucleation can be extremely low, and it may take googles of years for our neighborhood to be hit by a bubble wall. In scalar field models, the time of apocalypse depends on the slope of the energy hill and may come as soon as in 20 billion years.
3
Exactly when this is going to happen is hard to predict. The rate of bubble nucleation can be extremely low, and it may take googles of years for our neighborhood to be hit by a bubble wall. In scalar field models, the time of apocalypse depends on the slope of the energy hill and may come as soon as in 20 billion years.
Fire in the Equations
What is it that breathes fire into the equations and makes a universe for them to describe?
—STEPHEN HAWKING
A
lfonso the Wise, the thirteenth-century king of Castile, had a great respect for astronomy. And for a very practical reason: the knowledge of precise locations of planets on the sky was vital for casting accurate horoscopes. To improve their accuracy, Alfonso commissioned new astronomical tables based on Ptolemy’s model of the universe—then the latest word in cosmology. But when the intricacies of Ptolemy’s system were explained to him, Alfonso was rather skeptical: “If the Lord Almighty had consulted me before embarking upon creation, I should have recommended something simpler.”
1
lfonso the Wise, the thirteenth-century king of Castile, had a great respect for astronomy. And for a very practical reason: the knowledge of precise locations of planets on the sky was vital for casting accurate horoscopes. To improve their accuracy, Alfonso commissioned new astronomical tables based on Ptolemy’s model of the universe—then the latest word in cosmology. But when the intricacies of Ptolemy’s system were explained to him, Alfonso was rather skeptical: “If the Lord Almighty had consulted me before embarking upon creation, I should have recommended something simpler.”
1
King Alfonso could have made a similar remark about the worldview that I have described in this book. It asserts the existence of an infinite ensemble of universes, each containing a tapestry of regions with different particle physics. Regions where intelligent creatures can exist are rare and are separated by enormous distances. Rarer still are regions that are completely identical to one another, and yet there is an infinity of such regions. How wasteful of space, matter, and universes!
However, the number of universes is not something to be unduly concerned about. The new worldview saves a more precious commodity: it greatly reduces the number of arbitrary assumptions we have to make about the universe. The best theory is the one that explains the world with the fewest and simplest assumptions.
Earlier cosmological models suggested a Creator meticulously designing and fine-tuning the universe. Every detail of particle physics, each constant of nature, and all the primordial ripples had to be set just right. One can imagine the volumes and volumes of specifications the Creator handed down to his assistants to complete the job! The new worldview evokes a different image of the Creator. After some thought, he comes up with a set of equations for the fundamental theory of nature. This initiates the process of runaway creation. No further instructions are needed: the theory describes quantum nucleation of universes from nothing, the process of eternal inflation, and the creation of regions with every possible type of particle physics, ad infinitum. Any specific member of this ensemble of universes is incredibly complicated and would take an enormous amount of information to describe. But the entire ensemble can be codified in a relatively simple set of equations.
2
2
How do we know which portrait of the Creator is closer to the truth? Did he strive to optimize the use of “resources” like space and matter, or was he more concerned about having a concise mathematical description of nature? Unfortunately, he does not give interviews, but his product—the universe—leaves little doubt about what kind of Creator he is.
A casual look at the universe shows that space and matter are wasted in it with great abandon. Countless galaxies are scattered over immense stretches of nearly empty space. The galaxies fall into a few different classes, like spiral and elliptical, dwarf and giant. But apart from that, they are very similar to one another. The Creator makes it very clear that he is not embarrassed to repeat himself endlessly.
A more detailed examination reveals that the Creator is obsessed with mathematics. Pythagoras, in the sixth century B.C., was probably the first to suggest that mathematical relations were at the heart of all physical phenomena.
His insight was confirmed by centuries of scientific research, and we now take it for granted that nature follows precise mathematical laws. But if you stop to think about it, this fact is highly peculiar.
His insight was confirmed by centuries of scientific research, and we now take it for granted that nature follows precise mathematical laws. But if you stop to think about it, this fact is highly peculiar.
Mathematics appears to be a product of pure thought, with a very loose relation to experience. But then how come it is so ideally suited to describing the physical universe? This is what the physicist Eugene Wigner called “the unreasonable effectiveness of mathematics in natural sciences.” Consider the
ellipse
as a simple example. It was known to the ancient Greeks as the curve you get by cutting a cone with a plane at an angle. Archimedes and other Greek mathematicians studied the properties of the ellipse out of sheer interest in geometry. Then, almost two thousand years later, Johannes Kepler discovered that planets in their motion around the Sun describe ellipses with a remarkable accuracy. But what does the motion of Venus and Mars have to do with sections of a cone?
ellipse
as a simple example. It was known to the ancient Greeks as the curve you get by cutting a cone with a plane at an angle. Archimedes and other Greek mathematicians studied the properties of the ellipse out of sheer interest in geometry. Then, almost two thousand years later, Johannes Kepler discovered that planets in their motion around the Sun describe ellipses with a remarkable accuracy. But what does the motion of Venus and Mars have to do with sections of a cone?
Closer to home, in the 1960s my mathematician friend Victor Kac investigated a class of intricate mathematical structures now known as Kac-Moody algebras. His only motivation was his nose, which told him that the structures smelled interesting and could yield some beautiful mathematics. No one could have predicted that in a couple of decades these algebras would play a major role in string theory.
These examples are not exceptions. More often than not, physicists discover that the mathematics they need to describe a new class of phenomena has already been studied by mathematicians, for reasons that have nothing to do with the phenomena in question. It appears that the Creator shares the mathematicians’ sense of beauty. Many physicists rely on his idiosyncrasy and use mathematical beauty as a guide in their search for new theories. According to Paul Dirac, one of the pioneers of quantum mechanics, “It is more important to have beauty in one’s equations than to have them fit experiment … because the discrepancy may be due to minor features … that will get cleared up with further development of the theory.”
3
3
Mathematical beauty is no easier to define than beauty in art.
4
An example of what mathematicians find beautiful is what is known as
Euler’s formula
,
e
iπ
+ 1 = 0. One criterion for beauty is simplicity, but simplicity alone does not do it. The relation 1 + 1 = 2 is simple, but not particularly beautiful because it is trivial. In contrast, Euler’s formula shows a rather surprising connection between three seemingly unrelated numbers: the number
e
,
which is related to “natural” logarithms; the “imaginary” number
i
—the square root of –1; and the number π—the ratio of the circumference of a circle to its diameter. We can call this property “depth.” Beautiful mathematics combines simplicity with depth.
5
4
An example of what mathematicians find beautiful is what is known as
Euler’s formula
,
e
iπ
+ 1 = 0. One criterion for beauty is simplicity, but simplicity alone does not do it. The relation 1 + 1 = 2 is simple, but not particularly beautiful because it is trivial. In contrast, Euler’s formula shows a rather surprising connection between three seemingly unrelated numbers: the number
e
,
which is related to “natural” logarithms; the “imaginary” number
i
—the square root of –1; and the number π—the ratio of the circumference of a circle to its diameter. We can call this property “depth.” Beautiful mathematics combines simplicity with depth.
5
If indeed the Creator has a mathematician’s mind, then the equations of the fundamental theory of nature should be wonderfully simple and unbelievably deep. Some people think that this final theory is the theory of strings, which we are now in the process of discovering. This theory is definitely very deep. It does not look simple, but simplicity may emerge when the theory is better understood.
If we ever discover the final theory of nature, the question will still remain: Why this theory? Mathematical beauty may be useful as a guide, but it is hard to imagine that it will suffice to select a unique theory out of the infinite number of possibilities. As the physicist Max Tegmark put it, “Why should one mathematical structure, and only one, out of all the countless mathematical structures, be endowed with physical existence?”
6
Tegmark, now at Massachusetts Institute of Technology, suggested a possible way out of this impasse.
6
Tegmark, now at Massachusetts Institute of Technology, suggested a possible way out of this impasse.
His proposal is as simple as it is radical: he argues that there should be a universe corresponding to each and every mathematical structure.
7
There is, for example, a Newtonian universe governed by the laws of Euclidean geometry, classical mechanics, and Newton’s theory of gravitation. There are also universes where space has an infinite number of dimensions, and others having two dimensions of time. Even harder to imagine is a universe governed by the algebra of quaternions,
bn
which has neither space nor time.
7
There is, for example, a Newtonian universe governed by the laws of Euclidean geometry, classical mechanics, and Newton’s theory of gravitation. There are also universes where space has an infinite number of dimensions, and others having two dimensions of time. Even harder to imagine is a universe governed by the algebra of quaternions,
bn
which has neither space nor time.
Tegmark asserts that all these universes exist “out there.” We are not aware of them, just as we are not aware of other universes nucleating out of nothing. The mathematical structures in some of the universes are intricate enough to allow the emergence of “self-aware substructures” like you and
me. Such universes are rare, but of course they are the only ones that can be observed.
me. Such universes are rare, but of course they are the only ones that can be observed.
We have no evidence to support this dramatic extension of reality. The only reason for elevating universes with other mathematical structures to the realm of existence is to avoid explaining why they do not exist. This may be enough to convince some philosophers, but physicists need something more substantial. In the spirit of the principle of mediocrity, one could try to show that the fundamental theory of our universe is in some sense typical of all the theories rich enough to harbor observers. This would lend support to Tegmark’s extended multiverse.
If successful, this line of reasoning would drive the Creator entirely out of the picture. Inflation relieved him of the job of setting up the initial conditions of the big bang, quantum cosmology unburdened him of the task of creating space and time and starting up inflation, and now he is being evicted from his last refuge—the choice of the fundamental theory of nature.
Tegmark’s proposal, however, faces a formidable problem. The number of mathematical structures increases with increasing complexity, suggesting that “typical” structures should be horrendously large and cumbersome. This seems to be in conflict with the simplicity and beauty of the theories describing our world.
8
It thus appears that the Creator’s job security is in no immediate danger.
8
It thus appears that the Creator’s job security is in no immediate danger.
Philosophers and theologians have been arguing for centuries, trying to decide whether the universe is finite or infinite, stationary or evolving, eternal or transient. You might have thought that all possible answers to these questions have already been anticipated. However, the worldview emerging from recent developments in cosmology is not what anyone had expected. Instead of choosing between conflicting options, it suggests that each of them has some element of truth.
At the heart of the new worldview is the picture of an eternally inflating universe. It consists of isolated “island universes,” where inflation has ended, immersed in the inflating sea of false vacuum. The boundaries of these postinflationary islands are rapidly expanding, but the gaps that separate
them are widening even faster. Thus there is always room for more island universes to form, and their number increases without bound.
them are widening even faster. Thus there is always room for more island universes to form, and their number increases without bound.
Viewed from the inside, each island is a self-contained infinite universe. We live in one of these island universes, and our observable region is one of the infinite number of O-regions that it contains. It is conceivable that billions of years from now our descendants will travel to other O-regions, but a voyage to another island universe is impossible, even in principle. No matter how long we travel and how fast, we are forever confined to our own island universe.
The entire eternally inflating spacetime originated as a minuscule closed universe. It tunneled, quantum-mechanically, out of nothing and immediately plunged into the never-ending fury of inflation. Thus the universe is eternal, but it did have a beginning.
Inflation rapidly blows the universe up to an enormous size, but from a global viewpoint it always remains closed and finite. And yet, due to the peculiar structure of inflationary spacetime, it contains an unlimited number of infinite island universes.
Constants of nature that shape the character of our world take different values in other island universes. Most of these universes are drastically different from ours, and only a tiny fraction of them are hospitable to life.
9
Observers in each habitable island will find that their universe evolves from a big bang to a big crunch. However, in the global view, all types of islands at all stages of their evolution are present simultaneously. This situation is analogous to that of the human population of the Earth. Each person starts as a baby and grows older with time, but the entire population includes people of all ages at any given moment. Although the total volume of the universe grows with time, the fraction of space occupied by each type of island does not change. In this sense, the eternally inflating universe is stationary.
9
Observers in each habitable island will find that their universe evolves from a big bang to a big crunch. However, in the global view, all types of islands at all stages of their evolution are present simultaneously. This situation is analogous to that of the human population of the Earth. Each person starts as a baby and grows older with time, but the entire population includes people of all ages at any given moment. Although the total volume of the universe grows with time, the fraction of space occupied by each type of island does not change. In this sense, the eternally inflating universe is stationary.
A striking feature of the new worldview is the existence of multiple “other worlds” beyond our observable region. Some of them are rather uncontroversial. Very few people, for example, would question the reality of other O-regions, even though they cannot be observed. We do have some circumstantial evidence for multiple island universes with diverse properties. As for the other, disconnected spacetimes that nucleated out of nothing, we have no idea how to test their existence observationally.
The picture of quantum tunneling from nothing raises another intriguing
question. The tunneling process is governed by the same fundamental laws that describe the subsequent evolution of the universe. It follows that the laws should be “there” even prior to the universe itself. Does this mean that the laws are not mere descriptions of reality and can have an independent existence of their own? In the absence of space, time, and matter, what tablets could they be written upon? The laws are expressed in the form of mathematical equations. If the medium of mathematics is the mind, does this mean that mind should predate the universe?
question. The tunneling process is governed by the same fundamental laws that describe the subsequent evolution of the universe. It follows that the laws should be “there” even prior to the universe itself. Does this mean that the laws are not mere descriptions of reality and can have an independent existence of their own? In the absence of space, time, and matter, what tablets could they be written upon? The laws are expressed in the form of mathematical equations. If the medium of mathematics is the mind, does this mean that mind should predate the universe?
This takes us far into the unknown, all the way to the abyss of great mystery. It is hard to imagine how we can ever get past this point. But as before, that may just reflect the limits of our imagination.
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