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Vilenkin: Whitrow Lecture The Principle of Mediocrity

For the 2011 RAS Whitrow Lecture, discusses the new cosmological worldview in which much of the is in a state of constant accelerated expansion, called inflation, and we live on one of the many bubbles with diverse properties that are constantly being formed. What is the origin of this worldview, its possible observational tests, and its implications for the beginning and the end of the universe?

he universe as we know it originated in picture – the structure of the universe beyond a great explosion that we call the Big The Whitrow Lectures our cosmic horizon. TBang. For nearly a century cosmologists The Whitrow Lectures were established The end of inflation is triggered by , studied the aftermath of this explosion: how the thanks to a legacy from Gerald Whitrow, probabilistic processes and does not occur every­ universe expanded and cooled down, how gal- who died in 2000. Prof. Whitrow’s career where at once. Regions where false axies were gradually pulled together by gravity, spanned , , and decays somewhat later are “rewarded” by a etc. The of the itself came into the history and philosophy of , and larger inflationary expansion, so false vacuum focus only relatively recently. It is the subject of he worked hard to ensure that science regions tend to multiply faster than they decay. the of inflation, which was developed reached as wide an audience as possible. In our cosmic neighbourhood inflation ended in the early 1980s by , The Whitrow Lectures cover topics 13.7 billion years ago, but it probably still con- and others, and has led to a radically new global in cosmology, usually with particular tinues in remote parts of the universe, and other picture of the universe. According to this new reference to philosophical aspects of the “normal” regions like ours are constantly being picture, remote regions beyond our horizon subject. They are normally given biennially. formed. This never-ending process is called eter- are strikingly different from what we observe nal inflation. The eternal nature of inflation is here and may even obey different laws of phys- not automatic, but it is very generic. Practically ics. Here I will discuss the origin of the new of inflation is constant, the prices would dou- all models of inflation that have been discussed worldview, its possible observational tests, and ble, say, every 10 years. Cosmic inflation is a so far predict . its implications for the beginning and the end lot faster than that: depending on the model, The details of are model- of the universe. the doubling time can be as short as 10–37 sec- dependent; here, I will focus on models where I will start with a brief review of the theory of onds. In about 330 doubling the size of the it occurs through bubble nucleation. The low- inflation. The key role in this theory is played by universe will grow by a factor of 10100. No mat- energy regions then appear as tiny microscopic a peculiar entity called “false vacuum”. Vacuum ter what its initial size is, the universe will very bubbles and immediately start to grow, at a is just empty , but according to modern quickly become huge. Because the false vacuum rate rapidly approaching the . particle it is very different from “noth- is unstable, it eventually decays, producing a hot The bubbles keep growing without bound; in ing”. It is a physical object, endowed with energy fireball, and that’s the end of inflation. The fire- the meantime they are driven apart by the infla- density and pressure, and can be in a number ball continues to expand by inertia and evolves tionary expansion, making room for more bub- of different states. Particle physicists refer to along the lines of standard Big Bang cosmology. bles to form. We live in one of the bubbles and these states as different vacua. The properties Decay of the false vacuum plays the role of the can observe only a small part of it. No and the types of elementary particles differ from Big Bang in this scenario. how fast we travel, we cannot catch up with the one vacuum to another. The gravitational force The theory of inflation explained some other- expanding boundaries of our bubble, so for all induced by a false vacuum is rather peculiar: it is wise mysterious features of the Big Bang, which practical purposes we live in a self-contained repulsive. The higher the energy of the vacuum, simply had to be postulated before. It explained bubble universe. An unlimited number of bub- the stronger is the repulsion. The word “false” the expansion of the universe (it is due to the bles will be formed in the course of eternal infla- refers to the fact that this kind of vacuum is repulsive gravity of the false vacuum), its high tion. (For a review of inflation, including eternal unstable. It decays into a low-energy vacuum temperature (due to the high of inflation, see, e.g. Guth and Kaiser 2005.) like ours, and the excess energy goes to produce the false vacuum), and its observed a hot fireball of particles and . I should (false vacuum is very homogeneous: apart from A metaphysical interlude emphasize that false vacua with these strange quantum fluctuations, it has a constant energy At this point I would like to mention a remark- properties were not invented for the purposes of density). The theory has also made a number able and, to my mind, somewhat disturbing inflation: their existence follows from particle of testable predictions. It predicted that on the consequence of this picture of the universe (Gar- physics and . largest observable scales the universe should be riga and Vilenkin 2001). Because the number The theory of inflation assumes that at some accurately described by flat, Euclidean geom- of bubble is unlimited, and each of early time in its history the universe was in the etry. It also predicted a nearly scale-invariant them expands without bound, they will con- state of a high-energy false vacuum. Why it spectrum of small Gaussian density perturba- tain an unbounded number of regions of the should be so is a good question, and I will have tions. These predictions have been spectacularly size of our horizon. In each of these regions, something to say about it later in this article. confirmed by observations. By now inflation has the initial conditions at the Big Bang are set by The repulsive gravitational force produced by become the leading cosmological paradigm. random quantum processes during inflation, that vacuum would then cause a super-fast, so all possible initial conditions will be realized exponential expansion of the universe. There Eternal inflation with some probability. is a characteristic time – the doubling time – in Now that the theory of inflation is supported Now, the key point is that the number of dis- which the size of the universe would double. by the data in our observable region, we should tinct states in which any such region can be is This is similar to economic inflation: if the rate have some trust in what it tells us about the big finite. How is this possible? I can, for example,

A&G • October 2011 • Vol. 52 5.1 Vilenkin: Whitrow Lecture move my chair by one centimetre, by half a cen- 10500 (Lerche et al. 1987, Bousso and Polchinski no-one is there to complain about that. (For a timetre, by a quarter of a centimetre, and so 2000). Each solution corresponds to a vacuum non-technical review of ideas, see on, and apparently I have an infinite number of with its own types of elementary particles and Vilenkin 2006, Susskind 2006, Greene 2011.) possible states right there – because I can move its own values for the constants of Nature. Some of my colleagues find the multiverse it by an infinite number of possible displace- Now combine this with the theory of infla- theory alarming. Any theory in physics stands ments which get smaller and smaller. However, tion. Wherever they occur in the universe, high- or falls depending on whether its predictions states that are too close to one another cannot energy vacua will drive exponential inflationary agree with the data. But how can we verify be distinguished, even in principle, due to the expansion. Transitions between different vacua the existence of other bubble universes? Some quantum uncertainty. So quantum mechanics will occur through bubble nucleation, so there distinguished cosmologists, such as Paul Stein- tells us that the number of distinct states (in a will be bubbles within bubbles within bubbles. hardt and George Ellis, have even argued that finite volume) is finite. The number of quan- Each bubble type has a certain probability of the multiverse theory is unscientific, because it tum states in our observable region has been forming in the inflating space. So, inevitably, cannot be tested, even in principle. estimated as N ~ exp(10122). This is an unimagi- an unlimited number of bubbles of all Surprisingly, observational tests of nably large number. But the important point is possible types will be formed in The the multiverse picture may in fact that it is not infinite. the course of eternal inflation. be possible. One possibility is Thus, we have a finite number of states occur- This picture of the universe, constants‘‘ of to look for observational sig- ring in an infinite number of regions. The or multiverse as it is called, Nature take a wide natures of bubble collisions. inevitable conclusion is that each state hav- explains the long-standing range of values, As it expands, our bubble ing a non-zero probability occurs an infinite mystery of why the con- varying from will occasionally collide with number of times. In particular, there is an infi- stants of Nature appear to be one bubble to other bubbles. In fact, it will nite number of identical to ours. This fine-tuned for the emergence experience an infinite number means that scores of your exact duplicates are of life (see, e.g. Linde 1990). I another of collisions in the course of now reading this sentence. There should also be will give you just one example: the ’’ its history. Each such collision regions with all possible variations. For example, neutron mass. In our universe neutrons will produce an imprint in the cosmic there are some regions where the name of your are slightly heavier than protons. An isolated microwave background (CMB) comprising a dog is different and others where dinosaurs still neutron decays into a proton, an electron, and round spot of higher or lower radiation inten- roam the . Now, you may be wondering an antineutrino, but neutrons bound in atomic sity (Aguirre and Johnson 2011, Kleban 2011). whether all this is happening at the same time. nuclei are stabilized by nuclear forces. Sup- The CMB polarization within the spots is also This question does not have a definite answer, pose now that we decrease the neutron mass predicted to have a characteristic pattern (Czech because time and simultaneity are not uniquely by 1%. Then neutrons become lighter than et al. 2011). A detection of a spot which has defined in General Relativity. If, for example, we protons, and this will allow protons to decay the predicted characteristics and stands signifi- use matter density as the time variable in a bub- into neutrons and lighter particles. As a result cantly above the background would provide ble universe, then at each moment of time the the atomic nuclei will lose their electric charge. direct evidence for the existence of other bubble bubble interior is an infinite hyperbolic space, So there will be nothing to keep electrons in universes. The search is now on (Feeney et al. and each of us has an infinite number of dupli- atoms, and they fly away. Thus, if the neutron 2011), but unfortunately there is no guarantee cates presently living in our bubble. mass is decreased by 1%, we end up in a uni- that a bubble collision has occurred within our Note that infinity of space (or time) is not by verse without atoms, and it is hard to imagine cosmic horizon. Hence, failure to find signa- itself sufficient to warrant these conclusions. how life that is anything like ours could exist tures of bubble collisions on the microwave sky We could, for example, have the same galaxy in such a place. cannot be regarded as evidence against eternal endlessly repeated in an infinite space. So we Now, if we increase the mass of the neutron inflation. need some “randomizer”, a stochastic mecha- by 1%, it becomes so massive that it decays even Another interesting possibility is that our nism that would pick initial states for different inside a nucleus, turning into a proton. The elec- bubble universe could have tunnelled out of regions from the set of all possible states. Even tric repulsion between protons will then tear the an inflating vacuum where some of our three then, the entire set may not be exhausted if the nucleus apart, and the only possible atom will spatial dimensions were compactified. One total number of states is infinite. So the finite- be a single proton combined with an electron, can then hope to detect some residual asym- ness of N is important for the argument. In the which is hydrogen. Once again, it is hard to see metry in the expansion rate or in the spectrum case of eternal inflation, the finiteness ofN and how life can be possible in a universe with no of CMB temperature fluctuations. In a simple the randomness of initial conditions are both chemical elements other than hydrogen. model with one initially compact dimension, guaranteed by quantum mechanics. The situation is similar with other constants. this effect appears too small to be detected If you change them by relatively small amounts, (Blanco-Pillado and Salem 2010, Graham et The multiverse you end up with a universe that is not fit for life. al. 2010), but in other models the situation may So far I have assumed that other bubble uni- This seems to suggest that the constants were be more favourable. verses are similar to ours in terms of their fine-tuned by a Creator, in order to make a bio- physical properties. But this does not have to friendly universe for us to live in – exactly what The principle of mediocrity be so. , which is at present our the advocates of intelligent design have been As in a criminal trial, in the absence of direct best candidate for the fundamental theory of telling us all along! evidence for the multiverse, one can look for Nature, admits an immense number of solutions The multiverse picture offers a different expla- indirect, or circumstantial evidence. The idea describing vacua with diverse physical proper- nation. The constants of Nature take a wide is to use our theoretical model of the multiverse ties. These solutions are characterized by dif- range of values, varying from one bubble to to predict the constants of Nature that we can ferent compactifications of extra dimensions, another. Intelligent observers exist only in those expect to measure in our local region. One by branes wrapped around extra dimensions rare bubbles in which, by pure chance, the con- selection criterion is the so-called anthropic in different ways, etc. The number of possi- stants happen to be just right for life to evolve. principle, first introduced by Brandon Carter bilities is combinatorial and can be as high as The rest of the multiverse remains barren, but (1974). There are many different formulations

5.2 A&G • October 2011 • Vol. 52 Vilenkin: Whitrow Lecture of this principle in the literature, but most peo- The principle of mediocrity makes an opposite The measure problem has been around for ple understand it as stating the obvious fact that claim: that we should assume ourselves to be nearly two decades. During this time, several we can expect to measure only such values of typical in any class that we belong to, unless different measures have been proposed and the constants that are consistent with the exist- there is some evidence to the contrary (Garriga their properties have been investigated (for a ence of life. This “principle”, however, is guar- and Vilenkin 2008). recent review see Freivogel 2011). This work anteed to be true, so it is not very useful for Actually, I am surprised that this issue is so has shown that some of the proposals lead to testing the theory. controversial, since one can easily convince one- paradoxes or to a conflict with the data and In order to make testable predictions, we have self that the principle of mediocrity provides a should therefore be discarded. For instance, the to take a somewhat different approach (Vilenkin winning betting strategy. I will illustrate this proper time measure performed rather poorly, 1995). We can use the theory to derive the prob- with a simple example. Imagine that as you while the measure is still in the run- ability distribution for the constants measured arrive to a meeting of the Royal Society, the ning. It is unlikely, however, that this kind of by a randomly picked observer in the multi- organizers put a white or black hat on you. They phenomenological analysis will yield a unique verse. Assuming that we are typical observers have removed all mirrors, so you don’t know the prescription for the measure. This suggests that – the assumption that I called the principle of colour of your hat. You notice though that 80% some important element may be missing in our mediocrity – we can then predict the expected of people around you wear white hats and 20% understanding of cosmic inflation. range of values for the constants in our bubble. wear black hats. There may or may not be some Some people feel the problem is so grave that The width of this range will depend on the con- system as to how the hats are distributed. For it puts the validity of the theory of inflation fidence level at which we want to make the pre- example, the colour could be correlated with seriously in doubt (e.g. Steinhardt 2011). But diction. For example, if the desired confidence your sex, age, height, etc, but you don’t know. this is the view of only a small minority of cos- level is 95%, we should discard 2.5% at both Now, in order to register for the meeting, you mologists. Personally, I think the situation with tails of the distribution. Similar ideas have been have to bet £100 on the colour of your hat. How the theory of inflation is similar to that with suggested by Gott (1993), Leslie (1989), Page are you going to bet? One strategy is to assume Darwin’s theory of evolution some 100 years (1996) and Bostrom (2002); in fact, Carter’s that you are typical among the participants and ago. Both greatly expanded the range own interpretation of the bet that your hat is white. Another approach is of scientific inquiry, proposing an explanation was close to the principle of mediocrity. to say that you don’t really know whether you for something that was previously believed This strategy has been applied to the energy are typical or not. Then you throw a coin and impossible to explain. In both cases, the expla- density of our vacuum ρν, also known as “dark bet at random. With the first choice, 80% of nation was compelling, and no viable alterna- energy”. Steven Weinberg (1987) (see also Linde people will win, while with the second choice tives have been suggested. Darwin’s theory was

1987) has noted that in regions where ρν is large, only 50% win. Clearly, the principle of medi- widely accepted, even though some important it causes the universe to expand very fast, pre- ocrity provides a better betting strategy. With aspects remained unclear before the discovery venting matter from clumping into galaxies and more information, you can improve your odds of the genetic code. The theory of inflation may stars. Observers are not likely to evolve in such by narrowing your reference class accordingly. be similarly incomplete and may require addi- regions. Values of ρν much smaller than needed For instance, if you are a woman and you notice tional new ideas. But it also has a similar air of for galaxy formation require unnecessary fine- that most women wear black hats, you should inevitability. tuning and are also rather unlikely. Calculations bet that your hat is black. showed that most galaxies (and therefore most Beginning and end of the universe observers) are in regions where the The measure problem If inflation has no end, could it also have no density is about the same as the density of mat- A more serious challenge for the theory of the beginning? This would allow us to avoid many ter at the epoch of galaxy formation. The pre- multiverse is the so-called measure problem. As perplexing questions associated with the begin- diction is therefore that a similar value should discussed above, any event having a non-zero ning of the universe. Once you have a universe, be observed in our part of the universe (for a probability will happen in the course of eternal its evolution is described by the laws of phys- review and references see Vilenkin 2007). inflation, and it will happen an infinite number ics, but how do you describe the beginning? For the most part, physicists did not take these of times. Statistical predictions are based on rel- What caused the universe to appear? And who ideas seriously, but much to their surprise, dark ative frequencies of events in the limit of t → ∞. sets the initial conditions for the universe? It energy of roughly the expected magnitude was One finds, however, that the outcome sensitively would be an attractive solution if we could say detected in astronomical observations in the depends on the limiting procedure. More pre- that the universe has always been in the state late 1990s. As of now, there are no alternative cisely, it depends on what variable we use as of eternal inflation, without a beginning and explanations for the observed value of ρν. This time t. One possible choice is the “proper time” without end. may be our first evidence that there is indeed a measured by the clocks of co-moving observers. This idea, however, runs into an unexpected huge multiverse out there. It has changed many Another natural choice is the expansion factor obstacle. Arvind Borde, Alan Guth and I (2003) minds. (or scale factor) of the universe. The crux of have proved a theorem, which says that even the problem is that the volume of an inflating though inflation is eternal to the future, it can- Are we really typical? universe grows exponentially with time, and the not be eternal to the past. More precisely, the The principle of mediocrity has been the sub- numbers of all kinds of events grow accordingly. theorem says that all geodesics in an inflation- ject of much controversy. It asserts that we are As a result, most of the events will always be ary , except a set of measure zero, are typical observers, but there will always be some near the cutoff time, so it is not surprising that incomplete to the past. This means that inflation unfortunate creatures in the multiverse who will the resulting probability measure depends on must have had some sort of beginning. We are measure atypical values of the constants. How exactly how the cutoff is introduced. then faced with the question of what happened can we be sure that we are not them? Hartle On the positive side, predictions for the CMB before inflation. And whatever the answer is, we and Srednicki (2007) have argued, for exam- multipoles and for the dark energy are not very can keep asking: “And what happened before ple, that we should never assume ourselves to sensitive to the choice of measure. But as a mat- that?” So it seems that one of the most basic be typical in some class of observers, unless ter of principle, the theory will remain incom- questions of cosmology – What was the begin- we have evidence to back up that assumption. plete until the measure is fully specified. ning of the universe? – does not have a satisfac-

A&G • October 2011 • Vol. 52 5.3 Vilenkin: Whitrow Lecture tory answer. bourhood will be engulfed by a negative-energy with m ~ 1 eV (e.g. Hamann et al. 2010). The only way around this problem of infinite bubble. The expansion will then locally turn A major unresolved problem of the inflationary regress that has been suggested so far is the idea into contraction, and our region will collapse cosmology is the measure problem. Its resolu- that the universe could be spontaneously created to a . The bubble will arrive without tion may require radically new ideas. One possi- out of nothing. We often hear that nothing can warning, since it expands at nearly the speed of bility that has been recently suggested (Garriga come out of nothing. Indeed, matter has positive light. In fact, it may be rushing towards us at and Vilenkin 2009) is that the dynamics of energy, and demands that this very moment. the inflationary multiverse has a dual, “holo- any initial state should have the same energy. graphic” description, in the form of a quantum However, it is a mathematical fact that the Outlook theory defined at the future boundary of energy of a closed universe is equal to zero. In In summary, I have described the new world- spacetime. The measure of the multiverse can such a universe, the positive energy of matter view that has emerged from inflationary cos- then be related to the short-distance cutoff in is exactly compensated by the negative energy mology. According to this view, inflation is a that theory. This and other possibilities are now of the gravitational field, so the total energy is never-ending process, constantly producing new being explored. ● zero. Another conserved quantity is the electric “bubble universes” with diverse properties. This charge, but once again it turns out that the total multiverse picture can be tested both by direct Alexander Vilenkin, Institute of Cosmology, charge must vanish in a closed universe. This is observation of bubble collisions and indirectly, Dept of Physics and , Tufts University, not difficult to understand. Suppose the uni- using the principle of mediocrity. The prediction Massachusetts, USA. This lecture was originally verse has the form of a 3D sphere, and imagine for the dark energy based on this principle has delivered at the RAS National Astronomy Meeting placing a positive charge at the “south pole” of already been confirmed. Here I will mention on 19 April 2011. that sphere. The lines of force emanating from some other observational tests that have been Acknowledgment. This work was supported in the charge will then converge at the north pole, suggested in the literature. part by grant PHY-0855447 from the National indicating that there must be an equal negative A potentially testable feature of bubble uni- Science Foundation. charge there. Thus, you cannot add an electric verses is their negative spatial curvature. The References charge anywhere in a closed universe without curvature parameter Ωk is different for different adding an opposite charge someplace else. bubbles, depending on the amount of inflation Aguirre A and Johnson M 2011 Rept. Prog. Phys. 74 074901. If all the conserved numbers of a closed uni- in the bubble interior. The probability distribu- Blanco-Pillado J J and Salem M P 2010 JCAP 1007 7. verse are equal to zero, then there is nothing tion for Ωk has been studied by Freivogel et al. Borde A et al. 2003 Phys. Rev. Lett. 90 151301. to prevent such a universe from being sponta- (2006) and by De Simone and Salem (2010). Bostrom N 2002 Anthropic Bias: Observational Selec- neously created out of nothing. In quantum They found that the detectable range of values tion Effects (Routledge, New York). –4 Bousso R and Polchinski J 2000 JHEP 6 6. mechanics, any process which is not strictly for the curvature (|Ωk| ≲ 10 ) has a non-negli- Carter B 1974 in Confrontation of Cosmological Theo- forbidden by the conservation laws will hap- gible probability, but at the same time the broad ries with the Data ed. Longair M S (Reidel, Dordrecht). pen with some probability. tail of the distribution extends to values that Czech B et al. 2010 JCAP 12 23. The newly born universes can have a variety are too small to be detected. Apart from the De Simone A and Salem M P 2010 Phys. Rev. D 81 083527. of sizes and can be filled with different types curvature, quantum fluctuations in the par- Feeney S et al. 2011 arXiv:1012.3667 [hep-th]. of vacua. Analysis shows that the most prob- ent vacuum may also produce a characteristic Freivogel B et al. 2006 JHEP 603 039. able universes are the ones having the smallest feature in the spectrum of gravitational waves Freivogel B 2011 arXiv:1105.0244 [hep-th]. initial size and the highest (for inside the bubble. Detection of either of these Garriga J and Vilenkin A 2001 Phys. Rev. D 64 023507. Garriga J and Vilenkin A 2008 Phys. Rev. D 77 043526. more discussion see Vilenkin 2006, chapter 17). effects would provide additional evidence for Garriga J and Vilenkin A 2009 JCAP 901 21. Once the universe is formed, it starts expand- eternal inflation. Gott J R 1993 Nature 363 315–319. ing rapidly, because of the high energy of the The principle of mediocrity has also been Graham P W et al. 2010 Phys. Rev. D 82 063524. vacuum. This provides the start of the scenario applied to explain the amount of Green B 2011 The Hidden Reality: Parallel Universes of eternal inflation. in the universe. The composition of dark mat- and the Hidden Laws of the Cosmos (Knopf). Guth A H and Kaiser D I 2005 Science 307 884–890. You can ask: “What caused the universe to ter is unknown, and one of the best motivated Hamann et al. 2010 Phys. Rev. Lett. 105 181301. pop out of nothing?” Surprisingly, no cause is hypotheses is that it is made up of very light Hartle J and Srednicki M 2007 Phys. Rev. D 75 needed. If you have a radioactive atom, it will particles called . The density of axionic 123 523. decay, and quantum mechanics gives the decay dark matter is set by quantum fluctuations dur- Kleban M 2011 arXiv:1107.2593 [hep-th]. Lerche W et al. 1987 Nucl. Phys. B 287 477. probability in a given interval of time. But if ing inflation and varies from one place in the Leslie J 1989 Universes (Routledge, New York). you ask why the atom decayed at this particular universe to another. Its value affects the forma- Linde A 1987 in 300 Years of Gravitation ed. Hawk- moment and not the other, the answer is that tion of galaxies; hence, there is an anthropic ing S W and Israel W (Cambridge University Press, there is no cause: the process is completely ran- selection effect. Tegmark et al. (2006) (see Cambridge). Linde A 1988 Phys. Lett. B 201 437. dom. Similarly, no cause is needed for quantum Linde 1988 for earlier work) have calculated the Linde A D 1990 and Inflationary Cos- creation of the universe. resulting probability distribution and found that mology (Harwood Academic, Chur). I would like to close this section with some the observed value of the dark matter density is Page D N 1996 Int. J. Mod. Phys. D5 583. important news about the end of the world. It close to the peak of the distribution. If indeed Pogosian L et al. 2004 JCAP 407 5. Steinhardt P 2011 Scientific American April. is often said that if the dark energy is a cosmo- the dark matter turns out to be axionic, this can Susskind L 2006 The Cosmic Landscape: String Theory logical constant, then the universe will continue be counted as a success of the theory. and the Illusion of Intelligent Design (Back Bay Books). expanding forever. This is true for our bubble Multiverse predictions for the neutrino masses Tegmark M et al. 2005 Phys. Rev. D 71 103 523. universe as a whole, but not for our local region. have been worked out by Tegmark et al. (2005) Tegmark M et al. 2006 Phys. Rev. D 73 023505. Vilenkin A 1995 74 846–849. In the multiverse picture, there must be a large and Pogosian et al. (2004), with the conclusion Phys. Rev. Lett. Vilenkin A 2006 Many Worlds in One: the Search for number of negative-energy vacua, and bubbles that the sum of neutrino masses should be ~1 eV. Other Universes (Hill and Wang, New York). of such vacua will inevitably form within our It is intriguing to note that recent neutrino oscil- Vilenkin A 2007 in Universe or Multiverse ed. by Carr B (nearly zero-energy) vacuum. At some point, lation experiments, as well as cosmological J (Cambridge University Press, Cambridge). probably in a very distant future, our neigh- data, point to the existence of sterile neutrinos Weinberg S 1987 Phys. Rev. Lett. 59 2607.

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