Sometimes science does something else. Sometimes it challenges us to reexamine our views of science itself. The usual centuries-old scientific framework envisions that when describing a physical system, a physicist needs to specify three things. We've seen all three in various contexts, but it's useful to gather them together here. First are the mathematical equations describing the relevant physical laws (for example, these might be Newton's laws of motion, Maxwell's equations of electricity and magnetism, or Schrodinger's equation of quantum mechanics). Second are the numerical values of all constants of nature that appear in the mathematical equations (for example, the constants determining the intrinsic strength of gravity and the electromagnetic forces or those determining the ma.s.ses of the fundamental particles). Third, the physicist must specify the system's "initial conditions" (such as a baseball being hit from home plate at a particular speed in a particular direction, or an electron starting out with a 50 percent probability of being found at Grant's Tomb and an equal probability of being found at Strawberry Fields). The equations then determine what things will be like at any subsequent time. Both cla.s.sical and quantum physics subscribe to this framework; they differ only in that cla.s.sical physics purports to tell us how things will definitely be at a given moment, while quantum physics provides the probability that things will be one way or another.

When it comes to predicting where a batted ball will land, or how an electron will move through a computer chip (or a model Manhattan), this three-step process is demonstrably powerful. Yet, when it comes to describing the totality of reality, the three steps invite us to ask deeper questions: Can we explain the initial conditions-how things were at some purportedly earliest moment? Can we explain the values of the constants-the particle ma.s.ses, force strengths, and so on-on which those laws depend? Can we explain why a particular set of mathematical equations describes one or another aspect of the physical universe?

The various multiverse proposals we've discussed have the potential to profoundly shift our thinking on these questions. In the Quilted Multiverse, the physical laws across the const.i.tuent universes are the same, but the particle arrangements differ; different particle arrangements now reflect different initial conditions in the past. In this multiverse, therefore, our perspective on the question of why the initial conditions in our universe were one way or another shifts. Initial conditions can and generally will vary from universe to universe, so there is no fundamental explanation for any particular arrangement. Asking for such an explanation is asking the wrong kind of question; it's invoking single-universe mentality in a multiverse setting. Instead, the question we should ask is whether somewhere in the multiverse is a universe whose particle arrangement, and hence initial conditions, agrees with what we see here. Better still, can we show that such universes abound? If so, the deep question of initial conditions would be explained with a shrug of the shoulders; in such a multiverse, the initial conditions of our universe would be in no more need of an explanation than the fact that somewhere in New York is a shoe store that carries your size.

In the inflationary multiverse, the "constants" of nature can and generally will vary from bubble universe to bubble universe. Recall from Chapter 3 Chapter 3 that environmental differences-the different Higgs field values permeating each bubble-give rise to different particle ma.s.ses and force properties. The same holds true in the Brane Multiverse, the Cyclic Multiverse, and the Landscape Multiverse, where the form of string theory's extra dimensions, together with various differences in fields and fluxes, result in universes with different features-from the electron's ma.s.s to whether there even is an electron to the strength of electromagnetism to whether there is an electromagnetic force to the value of the cosmological constant, and so on. In the context of these multiverses, asking for an explanation of the particle and force properties we measure is once again asking the wrong kind of question; it's a question borne of single-universe thinking. Instead, we should ask whether in any of these multiverses there's a universe with the physical properties we measure. Better would be to show that universes with our physical features are abundant, or at least are abundant among all those universes that support life as we know it. But as much as it's meaningless to ask for that environmental differences-the different Higgs field values permeating each bubble-give rise to different particle ma.s.ses and force properties. The same holds true in the Brane Multiverse, the Cyclic Multiverse, and the Landscape Multiverse, where the form of string theory's extra dimensions, together with various differences in fields and fluxes, result in universes with different features-from the electron's ma.s.s to whether there even is an electron to the strength of electromagnetism to whether there is an electromagnetic force to the value of the cosmological constant, and so on. In the context of these multiverses, asking for an explanation of the particle and force properties we measure is once again asking the wrong kind of question; it's a question borne of single-universe thinking. Instead, we should ask whether in any of these multiverses there's a universe with the physical properties we measure. Better would be to show that universes with our physical features are abundant, or at least are abundant among all those universes that support life as we know it. But as much as it's meaningless to ask for the the word with which Shakespeare wrote word with which Shakespeare wrote Macbeth Macbeth, so it's meaningless to ask the equations to pick out the the values of the particular physical features we see here. values of the particular physical features we see here.

The Simulated and Ultimate Multiverses are horses of a different color; they don't emerge from particular physical theories. Yet, they too have the potential to shift the nature of our questions. In these multiverses, the mathematical laws governing the individual universes vary. Thus, much as with varying initial conditions and constants of nature, varying laws suggest that it's as misguided to ask for an explanation of the particular laws in operation here. Different universes have different laws; we experience the ones we do because these are among the laws compatible with our existence.

Collectively, we see that the multiverse proposals summarized in Table 11.1 Table 11.1 render prosaic three primary aspects of the standard scientific framework that in a single-universe setting are deeply mysterious. In various multiverses, the initial conditions, the constants of nature, and even the mathematical laws are no longer in need of explanation. render prosaic three primary aspects of the standard scientific framework that in a single-universe setting are deeply mysterious. In various multiverses, the initial conditions, the constants of nature, and even the mathematical laws are no longer in need of explanation.

Should We Believe Mathematics?

n.o.bel laureate Steven Weinberg once wrote, "Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world."1 Weinberg was referring to the pioneering results of Ralph Alpher, Robert Herman, and George Gamow on the cosmic microwave background radiation, which I described in Weinberg was referring to the pioneering results of Ralph Alpher, Robert Herman, and George Gamow on the cosmic microwave background radiation, which I described in Chapter 3 Chapter 3. Although the predicted radiation is a direct consequence of general relativity combined with basic cosmological physics, it rose to prominence only after being discovered theoretically twice, a dozen years apart, and then being observed through a benevolent act of serendipity.

To be sure, Weinberg's remark has to be applied with care. Although his his desk has played host to an inordinate amount of mathematics that has proved relevant to the real world, far from every equation with which we theorists tinker rises to that level. In the absence of compelling experimental or observational results, deciding which mathematics should be taken seriously is as much art as it is science. desk has played host to an inordinate amount of mathematics that has proved relevant to the real world, far from every equation with which we theorists tinker rises to that level. In the absence of compelling experimental or observational results, deciding which mathematics should be taken seriously is as much art as it is science.

Indeed, this issue is central to all we've discussed in this book; it has also informed the book's t.i.tle. The breadth of multiverse proposals in Table 11.1 Table 11.1 might suggest a panorama of hidden realities. But I've t.i.tled this book in the singular to reflect the unique and uniquely powerful theme that underlies them all: the capacity of mathematics to reveal secreted truths about the workings of the world. Centuries of discovery have made this abundantly evident; monumental upheavals in physics have emerged time and again from vigorously following mathematics' lead. Einstein's own complex dance with mathematics provides a revealing case study. might suggest a panorama of hidden realities. But I've t.i.tled this book in the singular to reflect the unique and uniquely powerful theme that underlies them all: the capacity of mathematics to reveal secreted truths about the workings of the world. Centuries of discovery have made this abundantly evident; monumental upheavals in physics have emerged time and again from vigorously following mathematics' lead. Einstein's own complex dance with mathematics provides a revealing case study.

In the late 1800s when James Clerk Maxwell realized that light was an electromagnetic wave, his equations showed that light's speed should be about 300,000 kilometers per second-close to the value experimenters had measured. A nagging loose end was that his equations left unanswered the question: 300,000 kilometers per second relative to what? Scientists pursued the makeshift resolution that an invisible substance permeating s.p.a.ce, the "aether," provided the unseen standard of rest. But in the early twentieth century, Einstein argued that scientists needed to take Maxwell's equations more seriously. If Maxwell's equations didn't refer to a standard of rest, then there was no need for a standard of rest; light's speed, Einstein forcefully declared, is 300,000 kilometers per second relative to anything anything. Although the details are of historical interest, I'm describing this episode for the larger point: everyone had access to Maxwell's mathematics, but it took the genius of Einstein to embrace the mathematics fully. And with that move, Einstein broke through to the special theory of relativity, overturning centuries of thought regarding s.p.a.ce, time, matter, and energy.

During the next decade, in the course of developing the general theory of relativity, Einstein became intimately familiar with vast areas of mathematics that most physicists of his day knew little or nothing about. As he groped toward general relativity's final equations, Einstein displayed a master's skill in molding these mathematical constructs with the firm hand of physical intuition. A few years later, when he received the good news that observations of the 1919 solar eclipse confirmed general relativity's prediction that star light should travel along curved trajectories, Einstein confidently noted that had the results been different, "he would have been sorry for the dear Lord, since the theory is correct." I'm sure that convincing data contravening general relativity would have changed Einstein's tune, but the remark captures well how a set of mathematical equations, through their sleek internal logic, their intrinsic beauty, and their potential for wide-ranging applicability, can seemingly radiate reality.

Nevertheless, there was a limit to how far Einstein was willing to follow his own mathematics. Einstein did not take the general theory of relativity "seriously enough" to believe its prediction of black holes, or its prediction that the universe was expanding. As we've seen, others, including Friedmann, Lemaitre, and Schwarzschild, embraced Einstein's equations more fully than he, and their achievements have set the course of cosmological understanding for nearly a century. By contrast, during the last twenty or so years of his life, Einstein threw himself into mathematical investigations, pa.s.sionately striving for the prized achievement of a unified theory of physics. In a.s.sessing this work based on what we know now, one can't help but conclude that during those years Einstein was too too heavily guided-some might say blinded-by the thicket of equations with which he was constantly surrounded. And so, even Einstein, at various times in his life, made the wrong decision regarding which equations to take seriously and which to not. heavily guided-some might say blinded-by the thicket of equations with which he was constantly surrounded. And so, even Einstein, at various times in his life, made the wrong decision regarding which equations to take seriously and which to not.

The third revolution in modern theoretical physics, quantum mechanics, provides another case study, one of direct relevance to the story I've told in this book. Schrodinger wrote down his equation for how quantum waves evolve in 1926. For decades, the equation was viewed as relevant only to the domain of small things: molecules, atoms, and particles. But in 1957, Hugh Everett echoed Einstein's Maxwellian charge of a half century earlier: take the math seriously take the math seriously. Everett argued that Schrodinger's equation should apply to everything because all things material, regardless of size, are made from molecules, atoms, and subatomic particles. And as we've seen, this led Everett to the Many Worlds approach to quantum mechanics and to the Quantum Multiverse. More than fifty years later, we still don't know if Everett's approach is right. But by taking the mathematics underlying quantum theory seriously-fully seriously-he may have discovered one of the most profound revelations of scientific exploration.

The other multiverse proposals similarly rely on a belief that mathematics is tightly st.i.tched into the fabric of reality. The Ultimate Multiverse takes this perspective to its furthermost incarnation; mathematics, according to the Ultimate Multiverse, is is reality. But even with their less panoptic view on the connection between mathematics and reality, the other multiverse theories in reality. But even with their less panoptic view on the connection between mathematics and reality, the other multiverse theories in Table 11.1 Table 11.1 owe their genesis to numbers and equations played with by theorists sitting at desks-and scribbling in notebooks, and writing on chalkboards, and programming computers. Whether invoking general relativity, quantum mechanics, string theory, or mathematical insight more broadly, the entries in owe their genesis to numbers and equations played with by theorists sitting at desks-and scribbling in notebooks, and writing on chalkboards, and programming computers. Whether invoking general relativity, quantum mechanics, string theory, or mathematical insight more broadly, the entries in Table 11.1 Table 11.1 arise only because we a.s.sume that mathematical theorizing can guide us toward hidden truths. Only time will tell if this a.s.sumption takes the underlying mathematical theories too seriously, or perhaps not seriously enough. arise only because we a.s.sume that mathematical theorizing can guide us toward hidden truths. Only time will tell if this a.s.sumption takes the underlying mathematical theories too seriously, or perhaps not seriously enough.

If some or all of the mathematics that's compelled us to think about parallel worlds proves relevant to reality, Einstein's famous query, asking whether the universe has the properties it does simply because no other universe is possible, would have a definitive answer: no. Our universe is not the only one possible. Its properties could have been different. And in many of the multiverse proposals, the properties of the other member universes would would be different. In turn, seeking a fundamental explanation for why certain things are the way they are would be pointless. Instead, statistical likelihood or plain happenstance would be firmly inserted in our understanding of a cosmos that would be profoundly vast. be different. In turn, seeking a fundamental explanation for why certain things are the way they are would be pointless. Instead, statistical likelihood or plain happenstance would be firmly inserted in our understanding of a cosmos that would be profoundly vast.

I don't know if this is how things will turn out. No one does. But it's only through fearless engagement that we can learn our own limits. It's only through the rational pursuit of theories, even those that whisk us into strange and unfamiliar domains, that we stand a chance of revealing the expanse of reality.

*Note, as in Chapter 7 Chapter 7, that an airtight observational refutation of inflation would require the theory's commitment to a procedure for comparing infinite cla.s.ses of universes-something it has not yet achieved. However, most pract.i.tioners would agree that if, say, the microwave background data had looked different from Figure 3.4 Figure 3.4, their confidence in inflation would have plummeted, even though, according to the theory, there's a bubble universe in the Inflationary Multiverse in which those data would hold.

Notes.

Chapter 1: The Bounds of Reality.

1. The possibility that our universe is a slab floating in a higher dimensional realm goes back to a paper by two renowned Russian physicists-"Do We Live Inside a Domain Wall?," V. A. Rubakov and M. E. Shaposhnikov, The possibility that our universe is a slab floating in a higher dimensional realm goes back to a paper by two renowned Russian physicists-"Do We Live Inside a Domain Wall?," V. A. Rubakov and M. E. Shaposhnikov, Physics Letters B Physics Letters B 125 (May 26, 1983): 136-and does not involve string theory. The version I'll focus on in Chapter 5 emerges from advances in string theory in the mid-1990s. 125 (May 26, 1983): 136-and does not involve string theory. The version I'll focus on in Chapter 5 emerges from advances in string theory in the mid-1990s.

Chapter 2: Endless Doppelgangers.

1. The quote comes from the March 1933 issue of The quote comes from the March 1933 issue of The Literary Digest The Literary Digest. It is worth noting that the precision of this quote has recently been questioned by the Danish historian of science Helge Kragh (see his Cosmology and Controversy Cosmology and Controversy, Princeton: Princeton University Press, 1999), who suggests it may be a reinterpretation of a Newsweek Newsweek report from earlier that year in which Einstein was referring to the origin of cosmic rays. What is certain, however, is that by this year Einstein had given up his belief that the universe was static and accepted the dynamic cosmology that emerged from his original equations of general relativity. report from earlier that year in which Einstein was referring to the origin of cosmic rays. What is certain, however, is that by this year Einstein had given up his belief that the universe was static and accepted the dynamic cosmology that emerged from his original equations of general relativity.

2. This law tells us the force of gravitational attraction, F, between two objects, given the ma.s.ses, m This law tells us the force of gravitational attraction, F, between two objects, given the ma.s.ses, m1 and m and m2, of each, and the distance, r, between them. Mathematically, the law reads: F=Gm F=Gm1m2/r2, where G G stands for Newton's constant-an experimentally measured number that specifies the intrinsic strength of the gravitational force. stands for Newton's constant-an experimentally measured number that specifies the intrinsic strength of the gravitational force.

3. For the mathematically inclined reader, Einstein's equations are For the mathematically inclined reader, Einstein's equations are R Ruv g guvR=8GTuv where where g guv is the metric on s.p.a.cetime, is the metric on s.p.a.cetime, R Ruv is the Ricci curvature tensor, is the Ricci curvature tensor, R R is the scalar curvature, is the scalar curvature, G G is Newton's constant, and is Newton's constant, and T Tuv is the energy-momentum tensor. is the energy-momentum tensor.

4. In the decades since this famous confirmation of general relativity, questions have been raised regarding the reliability of the results. For distant starlight grazing the sun to be visible, the observations had to be carried out during a solar eclipse; unfortunately, bad weather made it a challenge to take clear photographs of the solar eclipse of 1919. The question is whether Eddington and his collaborators might have been biased by foreknowledge of the result they were seeking, and so when they culled photographs deemed unreliable because of weather interference, they eliminated a disproportionate number containing data that appeared not to fit Einstein's theory. A recent and thorough study by Daniel Kennefick (see In the decades since this famous confirmation of general relativity, questions have been raised regarding the reliability of the results. For distant starlight grazing the sun to be visible, the observations had to be carried out during a solar eclipse; unfortunately, bad weather made it a challenge to take clear photographs of the solar eclipse of 1919. The question is whether Eddington and his collaborators might have been biased by foreknowledge of the result they were seeking, and so when they culled photographs deemed unreliable because of weather interference, they eliminated a disproportionate number containing data that appeared not to fit Einstein's theory. A recent and thorough study by Daniel Kennefick (see www.arxiv.org, paper arXiv:0709.0685, which, among other considerations, takes account of a modern reevaluation of the photograph plates taken in 1919) convincingly argues that the 1919 confirmation of general relativity is, indeed, reliable.

5. For the mathematically inclined reader, Einstein's equations of general relativity in this context reduce to For the mathematically inclined reader, Einstein's equations of general relativity in this context reduce to[image] . The variable a(t) is the scale factor of the universe-a number whose value, as the name indicates, sets the distance scale between objects (if the value of . The variable a(t) is the scale factor of the universe-a number whose value, as the name indicates, sets the distance scale between objects (if the value of a(t) a(t) at two different times differs, say, by a factor of 2, then the distance between any two particular galaxies would differ between those times by a factor of 2 as well), at two different times differs, say, by a factor of 2, then the distance between any two particular galaxies would differ between those times by a factor of 2 as well), G G is Newton's constant, is Newton's constant,[image] is the density of matter/energy, and is the density of matter/energy, and k k is a parameter whose value can be 1, 0, or -1 according to whether the shape of s.p.a.ce is spherical, Euclidean ("flat"), or hyperbolic. The form of this equation is usually credited to Alexander Friedmann and, as such, is called the Friedmann equation. is a parameter whose value can be 1, 0, or -1 according to whether the shape of s.p.a.ce is spherical, Euclidean ("flat"), or hyperbolic. The form of this equation is usually credited to Alexander Friedmann and, as such, is called the Friedmann equation.

6. The mathematically inclined reader should note two things. First, in general relativity we typically define coordinates that are themselves dependent on the matter s.p.a.ce contains: we use galaxies as the coordinate carriers (acting as if each galaxy has a particular set of coordinates "painted" on it-so-called co-moving coordinates). So, to even identify a specific region of s.p.a.ce, we usually make reference to the matter that occupies it. A more precise rephrasing of the text, then, would be: The region of s.p.a.ce containing a particular group of N galaxies at time t The mathematically inclined reader should note two things. First, in general relativity we typically define coordinates that are themselves dependent on the matter s.p.a.ce contains: we use galaxies as the coordinate carriers (acting as if each galaxy has a particular set of coordinates "painted" on it-so-called co-moving coordinates). So, to even identify a specific region of s.p.a.ce, we usually make reference to the matter that occupies it. A more precise rephrasing of the text, then, would be: The region of s.p.a.ce containing a particular group of N galaxies at time t1 will have a larger volume at a later time t will have a larger volume at a later time t2. Second, the intuitively sensible statement regarding the density of matter and energy changing when s.p.a.ce expands or contracts makes an implicit a.s.sumption regarding the equation of state for matter and energy. There are situations, and we will encounter one shortly, where s.p.a.ce can expand or contract while the density of a particular energy contribution-the energy density of the so-called cosmological constant-remains unchanged. Indeed, there are even more-exotic scenarios in which s.p.a.ce can expand while the density of energy increases increases. This can happen because, in certain circ.u.mstances, gravity can provide a source of energy. The important point of the paragraph is that in their original form the equations of general relativity are not compatible with a static universe.

7. Shortly we will see that Einstein abandoned his static universe when confronted by astronomical data showing that the universe is expanding. It is worth noting, though, that his misgivings about the static universe predated the data. The physicist Willem de Sitter pointed out to Einstein that his static universe was unstable: nudge it a bit bigger, and it would grow; nudge it a bit smaller, and it would shrink. Physicists shy away from solutions that require perfect, undisturbed conditions for them to persist. Shortly we will see that Einstein abandoned his static universe when confronted by astronomical data showing that the universe is expanding. It is worth noting, though, that his misgivings about the static universe predated the data. The physicist Willem de Sitter pointed out to Einstein that his static universe was unstable: nudge it a bit bigger, and it would grow; nudge it a bit smaller, and it would shrink. Physicists shy away from solutions that require perfect, undisturbed conditions for them to persist.

8. In the big bang model, the outward expansion of s.p.a.ce is viewed much like the upward motion of a tossed ball: attractive gravity pulls on the upward-moving ball and so slows its motion; similarly, attractive gravity pulls on the outward-moving galaxies and so slows their motion. In neither case does the ongoing motion require a repulsive force. However, you can still ask: Your arm launched the ball skyward, so what "launched" the spatial universe on its outward expansion? We will return to this question in In the big bang model, the outward expansion of s.p.a.ce is viewed much like the upward motion of a tossed ball: attractive gravity pulls on the upward-moving ball and so slows its motion; similarly, attractive gravity pulls on the outward-moving galaxies and so slows their motion. In neither case does the ongoing motion require a repulsive force. However, you can still ask: Your arm launched the ball skyward, so what "launched" the spatial universe on its outward expansion? We will return to this question in Chapter 3 Chapter 3, where we will see that modern theory posits a short burst of repulsive gravity, operating during the earliest moments of cosmic history. We will also see that more refined data has provided evidence that the expansion of s.p.a.ce is not not slowing over time, which has resulted in a surprising-and as later chapters will make clear-potentially profound resurrection of the cosmological constant. slowing over time, which has resulted in a surprising-and as later chapters will make clear-potentially profound resurrection of the cosmological constant.

The discovery of the spatial expansion was a turning point in modern cosmology. In addition to Hubble's contributions, the achievement relied on the work and insights of many others, including Vesto Slipher, Harlow Shapley, and Milton Humason.

9. A two-dimensional torus is usually depicted as a hollow doughnut. A two-step process shows that this picture agrees with the description provided in the text. When we declare that crossing the right edge of the screen brings you back to the left edge, that's tantamount to identifying the entire right edge with the left edge. Were the screen flexible (made of thin plastic, say) this identification could be made explicit by rolling the screen into a cylindrical shape and taping the right and left edges together. When we declare that crossing the upper edge brings you to the lower edge, that too is tantamount to identifying those edges. We can make this explicit by a second manipulation in which we bend the cylinder and tape the upper and lower circular edges together. The resulting shape has the usual doughnutlike appearance. A misleading aspect of these manipulations is that the surface of the doughnut looks curved; were it coated with reflective paint, your reflection would be distorted. This is an artifact of representing the torus as an object sitting within an ambient three-dimensional environment. Intrinsically, as a two-dimensional surface, the torus is not curved. It is flat, as is clear when it's represented as a flat video-game screen. That's why, in the text, I focus on the more fundamental description as a shape whose edges are identified in pairs. A two-dimensional torus is usually depicted as a hollow doughnut. A two-step process shows that this picture agrees with the description provided in the text. When we declare that crossing the right edge of the screen brings you back to the left edge, that's tantamount to identifying the entire right edge with the left edge. Were the screen flexible (made of thin plastic, say) this identification could be made explicit by rolling the screen into a cylindrical shape and taping the right and left edges together. When we declare that crossing the upper edge brings you to the lower edge, that too is tantamount to identifying those edges. We can make this explicit by a second manipulation in which we bend the cylinder and tape the upper and lower circular edges together. The resulting shape has the usual doughnutlike appearance. A misleading aspect of these manipulations is that the surface of the doughnut looks curved; were it coated with reflective paint, your reflection would be distorted. This is an artifact of representing the torus as an object sitting within an ambient three-dimensional environment. Intrinsically, as a two-dimensional surface, the torus is not curved. It is flat, as is clear when it's represented as a flat video-game screen. That's why, in the text, I focus on the more fundamental description as a shape whose edges are identified in pairs.

10. The mathematically inclined reader will note that by "judicious slicing and paring" I am referring to taking quotients of simply connected covering s.p.a.ces by various discrete isometry groups. The mathematically inclined reader will note that by "judicious slicing and paring" I am referring to taking quotients of simply connected covering s.p.a.ces by various discrete isometry groups.

11. The quoted amount is for the current era. In the early universe, the critical density was higher. The quoted amount is for the current era. In the early universe, the critical density was higher.

12. If the universe were static, light that had been traveling for the last 13.7 billion years and has only just reached us would indeed have been emitted from a distance of 13.7 billion light-years. In an expanding universe, the object that emitted the light has continued to recede during the billions of years the light was in transit. When we receive the light, the object is thus farther away-much farther-than 13.7 billion light-years. A straightforward calculation using general relativity shows that the object (a.s.suming it still exists and has been continually riding the swell of s.p.a.ce) would now be about 41 billion light-years away. This means that when we look out into s.p.a.ce we can, in principle, see light from sources that are now as far as roughly 41 billion light-years. In this sense, the observable universe has a diameter of about 82 billion light-years. The light from objects farther than this distance would not yet have had enough time to reach us and so are beyond our cosmic horizon. If the universe were static, light that had been traveling for the last 13.7 billion years and has only just reached us would indeed have been emitted from a distance of 13.7 billion light-years. In an expanding universe, the object that emitted the light has continued to recede during the billions of years the light was in transit. When we receive the light, the object is thus farther away-much farther-than 13.7 billion light-years. A straightforward calculation using general relativity shows that the object (a.s.suming it still exists and has been continually riding the swell of s.p.a.ce) would now be about 41 billion light-years away. This means that when we look out into s.p.a.ce we can, in principle, see light from sources that are now as far as roughly 41 billion light-years. In this sense, the observable universe has a diameter of about 82 billion light-years. The light from objects farther than this distance would not yet have had enough time to reach us and so are beyond our cosmic horizon.

13. In loose language, you can envision that because of quantum mechanics, particles always experience what I like to call "quantum jitter": a kind of inescapable random quantum vibration that renders the very notion of the particle having a definite position and speed (momentum) approximate. In this sense, changes to position/speed that are so small that they're on par with the quantum jitters are within the "noise" of quantum mechanics and hence are not meaningful. In loose language, you can envision that because of quantum mechanics, particles always experience what I like to call "quantum jitter": a kind of inescapable random quantum vibration that renders the very notion of the particle having a definite position and speed (momentum) approximate. In this sense, changes to position/speed that are so small that they're on par with the quantum jitters are within the "noise" of quantum mechanics and hence are not meaningful.

In more precise language, if you multiply the imprecision in the measurement of position by the imprecision in the measurement of momentum, the result-the uncertainty-is always larger than a number called Planck's constant Planck's constant, named after Max Planck, one of the pioneers of quantum physics. In particular, this implies that fine resolutions in measuring the position of a particle (small imprecision in position measurement) necessarily entail large uncertainty in the measurement of its momentum and, by a.s.sociation, its energy. Since energy is always limited, the resolution in position measurements is thus limited too.

Also note that we will always apply these concepts in a finite spatial domain-generally in regions the size of today's cosmic horizon (as in the next section). A finite-sized region, however large, implies a maximum uncertainty in position measurements. If a particle is a.s.sumed to be in a given region, the uncertainty of its position is surely no larger than the size of the region. Such a maximum uncertainty in position then entails, from the uncertainty principle, a minimum amount of uncertainty in momentum measurements-that is, limited resolution in momentum measurements. Together with the limited resolution in position measurements, we see the reduction from an infinite to a finite number of possible distinct configurations of a particle's position and speed.

You might still wonder about the barrier to building a device capable of measuring a particle's position with ever greater precision. It too is a matter of energy. As in the text, if you want to measure a particle's position with ever greater precision, you need to use an ever more refined probe. To determine whether a fly is in a room, you can turn on an ordinary, diffuse overhead light. To determine if an electron is in a cavity, you need to illuminate it with the sharp beam of a powerful laser. And to determine the electron's position with ever greater accuracy you need to make that laser ever more powerful. Now, when an ever more powerful laser zaps an electron, it imparts an ever greater disturbance to its velocity. Thus, the bottom line is that precision in determining particles' positions comes at the cost of huge changes in the particles' velocities-and hence huge changes in particle energies. If there's a limit to how much energy particles can have, as there always will be, there's a limit to how finely their positions can be resolved.

Limited energy in a limited spatial domain thus gives finite resolution on both position and velocity measurements.

14. The most direct way to make this calculation is by invoking a result I will describe in nontechnical terms in The most direct way to make this calculation is by invoking a result I will describe in nontechnical terms in Chapter 9 Chapter 9: the entropy of a black hole-the logarithm of the number of distinct quantum states-is proportional to its surface area measured in square Planck units. A black hole that fills our cosmic horizon would have a radius of about 1028 centimeters, or roughly 10 centimeters, or roughly 1061 Planck lengths. Its entropy would therefore be about 10 Planck lengths. Its entropy would therefore be about 10122 in square Planck units. Hence the total number of distinct states is roughly 10 raised to the power 10 in square Planck units. Hence the total number of distinct states is roughly 10 raised to the power 10122, or 1010122.

15. You might be wondering why I'm not also incorporating fields. As we will see, particles and fields are complementary languages-a field can be described in terms of the particles of which it's composed, much like an ocean wave can be described in terms of its const.i.tuent water molecules. The choice of using a particle or field language is largely one of convenience. You might be wondering why I'm not also incorporating fields. As we will see, particles and fields are complementary languages-a field can be described in terms of the particles of which it's composed, much like an ocean wave can be described in terms of its const.i.tuent water molecules. The choice of using a particle or field language is largely one of convenience.

16. The distance that light can travel in a given time interval depends sensitively on the rate at which s.p.a.ce expands. In later chapters we will encounter evidence that the rate of spatial expansion is accelerating. If so, there is a limit to how far light can travel through s.p.a.ce, even if we wait an arbitrarily long time. Distant regions of s.p.a.ce would be receding from us so quickly that light we emit could not reach them; similarly, light they emit could not reach us. This would mean that cosmic horizons-the portion of s.p.a.ce with which we can exchange light signals-would not grow in size indefinitely. (For the mathematically inclined reader, the essential formulae are in Chapter 6, The distance that light can travel in a given time interval depends sensitively on the rate at which s.p.a.ce expands. In later chapters we will encounter evidence that the rate of spatial expansion is accelerating. If so, there is a limit to how far light can travel through s.p.a.ce, even if we wait an arbitrarily long time. Distant regions of s.p.a.ce would be receding from us so quickly that light we emit could not reach them; similarly, light they emit could not reach us. This would mean that cosmic horizons-the portion of s.p.a.ce with which we can exchange light signals-would not grow in size indefinitely. (For the mathematically inclined reader, the essential formulae are in Chapter 6, note 7 note 7.) 17. G. Ellis and G. Bundrit studied duplicate realms in an infinite cla.s.sical universe; J. Garriga and A. the quantum context. G. Ellis and G. Bundrit studied duplicate realms in an infinite cla.s.sical universe; J. Garriga and A. the quantum context.

Chapter 3: Eternity and Infinity.

1. One point of departure from the earlier work was d.i.c.ke's perspective, which focused on the possibility of an oscillating universe that would repeatedly go through a series of cycles-big bang, expansion, contraction, big crunch, big bang again. In any given cycle there would be remnant radiation suffusing s.p.a.ce. One point of departure from the earlier work was d.i.c.ke's perspective, which focused on the possibility of an oscillating universe that would repeatedly go through a series of cycles-big bang, expansion, contraction, big crunch, big bang again. In any given cycle there would be remnant radiation suffusing s.p.a.ce.

2. It is worth noting that even though they don't have jet engines, galaxies generally do exhibit some motion above and beyond that arising from the expansion of s.p.a.ce-typically the result of large-scale intergalactic gravitational forces as well as the intrinsic motion of the swirling gas cloud from which stars in the galaxies formed. Such motion is called It is worth noting that even though they don't have jet engines, galaxies generally do exhibit some motion above and beyond that arising from the expansion of s.p.a.ce-typically the result of large-scale intergalactic gravitational forces as well as the intrinsic motion of the swirling gas cloud from which stars in the galaxies formed. Such motion is called peculiar velocity peculiar velocity and is generally small enough that it can be safely ignored for cosmological purposes. and is generally small enough that it can be safely ignored for cosmological purposes.

3. The horizon problem is subtle, and my description of inflationary cosmology's solution slightly nonstandard, so for the interested reader let me elaborate here in a little more detail. First the problem, again: Consider two regions in the night sky that are so distant from one another that they have never communicated. And to be concrete, let's say each region has an observer who controls a thermostat that sets his or her region's temperature. The observers want the two regions to have the same temperature, but because the observers have been unable to communicate, they don't know how to set their respective thermostats. The natural thought is that since billions of years ago the observers were much closer, it would have been easy for them, way back then, to have communicated and thus to have ensured the two regions had equal temperatures. However, as noted in the main text, in the standard big bang theory this reasoning fails. Here's more detail on why. In the standard big bang theory, the universe is expanding, but because of gravity's attractive pull, the The horizon problem is subtle, and my description of inflationary cosmology's solution slightly nonstandard, so for the interested reader let me elaborate here in a little more detail. First the problem, again: Consider two regions in the night sky that are so distant from one another that they have never communicated. And to be concrete, let's say each region has an observer who controls a thermostat that sets his or her region's temperature. The observers want the two regions to have the same temperature, but because the observers have been unable to communicate, they don't know how to set their respective thermostats. The natural thought is that since billions of years ago the observers were much closer, it would have been easy for them, way back then, to have communicated and thus to have ensured the two regions had equal temperatures. However, as noted in the main text, in the standard big bang theory this reasoning fails. Here's more detail on why. In the standard big bang theory, the universe is expanding, but because of gravity's attractive pull, the rate rate of expansion slows over time. It's much like what happens when you toss a ball in the air. During its ascent it first moves away from you quickly, but because of the tug of earth's gravity, it steadily slows. The slowing down of spatial expansion has a profound effect. I'll use the tossed ball a.n.a.logy to explain the essential idea. Imagine a ball that undergoes, say, a six second ascent. Since it initially travels quickly (as it leaves your hand), it might cover the first half of the journey in only two seconds, but due to its diminishing speed it takes four more seconds to cover the second half of the journey. At the halfway point in time, three seconds, it was thus of expansion slows over time. It's much like what happens when you toss a ball in the air. During its ascent it first moves away from you quickly, but because of the tug of earth's gravity, it steadily slows. The slowing down of spatial expansion has a profound effect. I'll use the tossed ball a.n.a.logy to explain the essential idea. Imagine a ball that undergoes, say, a six second ascent. Since it initially travels quickly (as it leaves your hand), it might cover the first half of the journey in only two seconds, but due to its diminishing speed it takes four more seconds to cover the second half of the journey. At the halfway point in time, three seconds, it was thus beyond beyond the halfway mark in distance. Similarly, with spatial expansion that slows over time: at the halfway point in cosmic history, our two observers would be separated by the halfway mark in distance. Similarly, with spatial expansion that slows over time: at the halfway point in cosmic history, our two observers would be separated by more more than half their current distance. Think about what this means. The two observers would be closer together, but they would find it harder-not easier-to have communicated. Signals one observer sends would have half the time to reach the other, but the distance the signals would need to traverse is than half their current distance. Think about what this means. The two observers would be closer together, but they would find it harder-not easier-to have communicated. Signals one observer sends would have half the time to reach the other, but the distance the signals would need to traverse is more more than half of what it is today. Being allotted half the time to communicate across more than half their current separation only makes communication more difficult. than half of what it is today. Being allotted half the time to communicate across more than half their current separation only makes communication more difficult.

The distance between objects is thus only one consideration when a.n.a.lyzing their ability to influence each other. The other essential consideration is the amount of time that's elapsed since the big bang, as this constrains how far any purported influence could have traveled. In the standard big bang, although everything was indeed closer in the past, the universe was also expanding more quickly, resulting in less time, proportionally speaking, for influences to be exerted.

The resolution offered by inflationary cosmology is to insert a phase in the earliest moments of cosmic history in which the expansion rate of s.p.a.ce doesn't decrease like the speed of the ball tossed upwards; instead, the spatial expansion starts out slow and then continually picks up speed: the expansion accelerates. By the same reasoning we just followed, at the halfway point of such an inflationary phase our two observers will be separated by less less than half their distance at the end of that phase. And being allotted half the time to communicate across less than half the distance means it is easier at earlier times for them to communicate. More generally, at ever earlier times, accelerated expansion means there is more time, proportionally speaking-not less-for influences to be exerted. This would have allowed today's distant regions to have easily communicated in the early universe, explaining the common temperature they now have. than half their distance at the end of that phase. And being allotted half the time to communicate across less than half the distance means it is easier at earlier times for them to communicate. More generally, at ever earlier times, accelerated expansion means there is more time, proportionally speaking-not less-for influences to be exerted. This would have allowed today's distant regions to have easily communicated in the early universe, explaining the common temperature they now have.

Because the accelerated expansion results in a much greater total spatial expansion of s.p.a.ce than in the standard big bang theory, the two regions would have been much much closer together at the onset of inflation than at a comparable moment in the standard big bang theory. This size disparity in the very early universe is an equivalent way of understanding why communication between the regions, which would have proved impossible in the standard big bang, can be easily accomplished in the inflationary theory. If at a given moment after the beginning, the distance between two regions is less, it is easier for them to exchange signals. closer together at the onset of inflation than at a comparable moment in the standard big bang theory. This size disparity in the very early universe is an equivalent way of understanding why communication between the regions, which would have proved impossible in the standard big bang, can be easily accomplished in the inflationary theory. If at a given moment after the beginning, the distance between two regions is less, it is easier for them to exchange signals.

Taking the expansion equations seriously to arbitrarily early times (and for definiteness, imagine that s.p.a.ce is spherically shaped), we also see that the two regions would have initially separated more quickly in the standard big bang than in the inflationary model: that's how they became so much farther apart in the standard big bang compared with their separation in the inflationary theory. In this sense, the inflationary framework involves a period of time during which the rate of separation between these regions is slower than in the usual big bang framework.

Often, in describing inflationary cosmology, the focus is solely on the fantastic increase in expansion speed over the conventional framework, not on a decrease in speed. The difference in description derives from which physical features between the two frameworks one compares. If one compares the trajectories of two regions of a given distance apart in the very early universe, then in the inflationary theory those regions separate much faster than in the standard big bang theory; by today they are also much farther apart in the inflationary theory than in the conventional big bang. But if one considers two regions of a given distance apart today (like the two regions on opposite sides of the night sky upon which we've been focused), the description I've given is relevant. Namely, at a given moment in time in the very early universe, those regions were much closer together, and had been moving apart much more slowly, in a theory that invokes inflationary expansion as compared with one that doesn't. The role of inflationary expansion is to make up for the slower start by then propelling those regions apart ever more quickly, ensuring that they arrive at the same location in the sky that they would have in the standard big bang theory.

A fuller treatment of the horizon problem would include a more detailed specification of the conditions from which the inflationary expansion emerges as well as the subsequent processes by which, for example, the cosmic microwave background radiation is produced. But this discussion highlights the essential distinction between accelerated and decelerated expansion.

4. Note that by squeezing the bag, you inject energy into it, and since both ma.s.s and energy give rise to the resulting gravitational warpage, the increase in weight will be partially due to the increase in energy. The point, however, is that the increase in pressure itself also contributes to the increase in weight. (Also note that to be precise, we should imagine doing this "experiment" in a vacuum chamber, so we don't need to consider the buoyant forces due to the air surrounding the bag.) For everyday examples the increase is tiny. However, in astrophysical settings the increase can be significant. In fact, it plays a role in understanding why, in certain situations, stars necessarily collapse to form black holes. Stars generally maintain their equilibrium through a balance between outward-pushing pressure, generated by nuclear processes in the star's core, and inward-pulling gravity, generated by the star's ma.s.s. As the star exhausts its nuclear fuel, the positive pressure decreases, causing the star to contract. This brings all its const.i.tuents closer together and so increases their gravitational attraction. To avoid further contraction, additional outward pressure (what is labeled positive pressure, as in the next paragraph in the text) is needed. But the additional positive pressure itself generates additional attractive gravity and thus makes the need for additional positive pressure all the more urgent. In certain situations, this leads to a spiraling instability and the very thing that the star usually relies upon to counteract the inward pull of gravity-positive pressure-contributes so strongly to that very inward pull that a complete gravitational collapse becomes unavoidable. The star will implode and form a black hole. Note that by squeezing the bag, you inject energy into it, and since both ma.s.s and energy give rise to the resulting gravitational warpage, the increase in weight will be partially due to the increase in energy. The point, however, is that the increase in pressure itself also contributes to the increase in weight. (Also note that to be precise, we should imagine doing this "experiment" in a vacuum chamber, so we don't need to consider the buoyant forces due to the air surrounding the bag.) For everyday examples the increase is tiny. However, in astrophysical settings the increase can be significant. In fact, it plays a role in understanding why, in certain situations, stars necessarily collapse to form black holes. Stars generally maintain their equilibrium through a balance between outward-pushing pressure, generated by nuclear processes in the star's core, and inward-pulling gravity, generated by the star's ma.s.s. As the star exhausts its nuclear fuel, the positive pressure decreases, causing the star to contract. This brings all its const.i.tuents closer together and so increases their gravitational attraction. To avoid further contraction, additional outward pressure (what is labeled positive pressure, as in the next paragraph in the text) is needed. But the additional positive pressure itself generates additional attractive gravity and thus makes the need for additional positive pressure all the more urgent. In certain situations, this leads to a spiraling instability and the very thing that the star usually relies upon to counteract the inward pull of gravity-positive pressure-contributes so strongly to that very inward pull that a complete gravitational collapse becomes unavoidable. The star will implode and form a black hole.

5. In the approach to inflation I have just described, there is no fundamental explanation for why the inflaton field's value would begin high up on the potential energy curve, nor why the potential energy curve would have the particular shape it has. These are a.s.sumptions the theory makes. Subsequent versions of inflation, most notably one developed by Andrei Linde called In the approach to inflation I have just described, there is no fundamental explanation for why the inflaton field's value would begin high up on the potential energy curve, nor why the potential energy curve would have the particular shape it has. These are a.s.sumptions the theory makes. Subsequent versions of inflation, most notably one developed by Andrei Linde called chaotic inflation chaotic inflation, find that a more "ordinary" potential energy curve (a parabolic shape with no flat section that emerges from the simplest mathematical equations for the potential energy) can also yield inflationary expansion. To initiate the inflationary expansion, the inflaton field's value needs to be high up on this potential energy curve too, but the enormously hot conditions expected in the early universe would naturally cause this to happen.

6. For the diligent reader, let me note one additional detail. The rapid expansion of s.p.a.ce in inflationary cosmology entails significant cooling (much as a rapid compression of s.p.a.ce, or of most anything, causes a surge in temperature). But as inflation comes to a close, the inflaton field oscillates around the minimum of its potential energy curve, transferring its energy to a bath of particles. The process is called "re-heating" because the particles so produced will have kinetic energy and thus can be characterized by a temperature. As s.p.a.ce then continues to undergo more ordinary (non-inflationary) big bang expansion, the temperature of the particle bath steadily decreases. The important point, though, is that the uniformity set down by inflation provides uniform conditions for these processes, and so results in uniform outcomes. For the diligent reader, let me note one additional detail. The rapid expansion of s.p.a.ce in inflationary cosmology entails significant cooling (much as a rapid compression of s.p.a.ce, or of most anything, causes a surge in temperature). But as inflation comes to a close, the inflaton field oscillates around the minimum of its potential energy curve, transferring its energy to a bath of particles. The process is called "re-heating" because the particles so produced will have kinetic energy and thus can be characterized by a temperature. As s.p.a.ce then continues to undergo more ordinary (non-inflationary) big bang expansion, the temperature of the particle bath steadily decreases. The important point, though, is that the uniformity set down by inflation provides uniform conditions for these processes, and so results in uniform outcomes.

7. Alan Guth was aware of the eternal nature of inflation; Paul Steinhardt wrote about its mathematical realization in certain contexts; Alexander Vilenkin brought it to light in the most general terms. Alan Guth was aware of the eternal nature of inflation; Paul Steinhardt wrote about its mathematical realization in certain contexts; Alexander Vilenkin brought it to light in the most general terms.

8. The value of the inflaton field determines the amount of energy and negative pressure it suffuses through s.p.a.ce. The larger the energy, the greater the expansion rate of s.p.a.ce. The rapid expansion of s.p.a.ce, in turn, has a back reaction on the inflaton field itself: the faster the expansion of s.p.a.ce, the more violently the inflaton field's value jitters. The value of the inflaton field determines the amount of energy and negative pressure it suffuses through s.p.a.ce. The larger the energy, the greater the expansion rate of s.p.a.ce. The rapid expansion of s.p.a.ce, in turn, has a back reaction on the inflaton field itself: the faster the expansion of s.p.a.ce, the more violently the inflaton field's value jitters.

9. Let me address a question that may have occurred to you, one we will return to in Let me address a question that may have occurred to you, one we will return to in Chapter 10 Chapter 10. As s.p.a.ce undergoes inflationary expansion, its overall energy increases: the greater the volume of s.p.a.ce filled with an inflaton field, the greater the total energy (if s.p.a.ce is infinitely large, energy is infinite too-in this case we should speak of the energy contained in a finite region of s.p.a.ce as the region grows larger). Which naturally leads one to ask: What is the source of this energy? For the a.n.a.logous situation with the champagne bottle, the source of additional energy in the bottle came from the force exerted by your muscles. What plays the role of your muscles in the expanding cosmos? The answer is gravity. Whereas your muscles were the agent that allowed the available s.p.a.ce inside the bottle to expand (by pulling out the cork), gravity is the agent that allows the available s.p.a.ce in the cosmos to expand. What's vital to realize is that the gravitational field's energy can be arbitrarily negative. Consider two particles falling toward each other under their mutual gravitational attraction. Gravity coaxes the particles to approach each other faster and faster, and as they do, their kinetic energy gets ever more positive. The gravitational field can supply the particles with such positive energy because gravity can draw down its own energy reserve, which becomes arbitrarily negative in the process: the closer the particles approach each other, the more negative the gravitational energy becomes (equivalently, the more positive the energy you'd need to inject to overcome the force of gravity and separate the particles once again). Gravity is thus like a bank that has a bottomless credit line and so can lend endless amounts of money; the gravitational field can supply endless amounts of energy because its own energy can become ever more negative. And that's the energy source that inflationary expansion taps.

10. I will use the term "bubble universe," although the imagery of a "pocket universe" that opens up within the ambient inflaton-filled environment is a good one too (that term was coined by Alan Guth). I will use the term "bubble universe," although the imagery of a "pocket universe" that opens up within the ambient inflaton-filled environment is a good one too (that term was coined by Alan Guth).

11. For the mathematically inclined reader, a more precise description of the horizontal axis in For the mathematically inclined reader, a more precise description of the horizontal axis in Figure 3.5 Figure 3.5 is as follows: consider the two-dimensional sphere comprising the points in s.p.a.ce at the time the cosmic microwave background photons began to stream freely. As with any two-sphere, a convenient set of coordinates on this locus are the angular coordinates from a spherical polar coordinate system. The temperature of the cosmic microwave background radiation can then be viewed as a function of these angular coordinates and, as such, can be decomposed in a Fourier series using as a basis the standard spherical harmonics, is as follows: consider the two-dimensional sphere comprising the points in s.p.a.ce at the time the cosmic microwave background photons began to stream freely. As with any two-sphere, a convenient set of coordinates on this locus are the angular coordinates from a spherical polar coordinate system. The temperature of the cosmic microwave background radiation can then be viewed as a function of these angular coordinates and, as such, can be decomposed in a Fourier series using as a basis the standard spherical harmonics,[image] . The vertical axis in . The vertical axis in Figure 3.5 Figure 3.5 is related to the size of the coefficients for each mode in this expansion-farther to the right on the horizontal axis corresponds to smaller angular separation. For technical details, see for example Scott Dodelson's excellent book is related to the size of the coefficients for each mode in this expansion-farther to the right on the horizontal axis corresponds to smaller angular separation. For technical details, see for example Scott Dodelson's excellent book Modern Cosmology Modern Cosmology (San Diego, Calif.: Academic Press, 2003). (San Diego, Calif.: Academic Press, 2003).

12. A little more precisely, it is not the strength of the gravitational field, per se, that determines the slowing of time, but rather the strength of the gravitational potential. For instance, if you were to hang out inside a spherical cavity at the center of a m