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Inflation and Cosmological Perturbations

This talk, which was presented at Stephen Hawking's 60th birthday conference, begins with a discussion of the early development of the theory of inflationary density perturbations. Stephen played a crucial role in this work, at every level. Much of the foundation for this work was laid by Stephen's 1966 paper on cosmological density perturbations, and by his 1977 paper with Gary Gibbons on quantum field theory in de Sitter space. Stephen was a major participant in the new work, and he was also a co-organizer of the 1982 Nuffield Workshop, where divergent ideas about inflationary density perturbations were thrashed about until a consensus emerged. In the second part of the talk I summarize the recent observational successes of these predictions, I present a graph of the probability distribution for the time of last scattering for CMB photons, and I summarize a recent theorem by Borde, Vilenkin, and me which shows that although inflation is generically eternal to the future, an inflationary region of spacetime must be incomplete in null and timelike past directions.

The three-week period of the Nuffield meeting was certainly among the most exciting times of my life. I will always cherish the memories of those events, and I will always be grateful to Stephen for the important role that he played in making them happen.
When "old inflation" [17] was first invented, we knew at once that we could identify at least one possible source of density perturbations: namely, the random nucleation of bubbles. In that model the phase transition was assumed to be strongly first order, so it ended with bubbles of the new phase nucleating and colliding, producing some kind of inhomogeneous froth similar to boiling water. But it was of course not clear immediately whether these density perturbations would have the right properties to serve as the seeds for structure formation. Stephen played an important role in sorting out this question, as one of the key papers was that of Hawking, Stewart, and Moss [29]. Once the issue was understood, the answer was clear. The density perturbations created in the old inflationary model were ridiculously large -colossal -so there was absolutely no hope of getting a universe from it that looked like ours.
Fortunately, inflation was resurrected by the invention of "new inflation" by Linde [33], and Albrecht and Steinhardt [2], and it was in that context that the question of density perturbations could first be intelligently addressed. For new inflation, however, the question of density perturbations was much more subtle.
The first I remember hearing about this was at a meeting at Moriond in March of 1982. I chatted with Michael Turner, and I guess he had been in touch with Jim Bardeen. As far as I know, Jim was one of first people to really worry about this issue. Jim had realized that in the context of the new inflationary model, any density perturbations present before inflation would be suppressed by an enormous exponential factor as they evolved through the inflationary regime. This meant that if the model was to have any chance of viability, some mechanism would have to be found to allow the density perturbations to survive. According to Michael, Jim was exploring the possibility that there might be some mechanism at the very end of inflation that would amplify the density perturbations which had been suppressed out of sight by the process of inflation itself. At about the same time I had a conversation with Dave Schramm, who was also aware of this problem, and who thought that maybe we would have to learn about turbulence to understand how density perturbations could survive inflation.
In May 1982, there was a chance meeting in Chicago of three physicists interested in these questions: Michael Turner, Paul Steinhardt, and Stephen Hawking. Michael was of course based in Chicago, as he still is, Paul was visiting to give a colloquium, and Stephen was by chance visiting at the same time to collaborate with Jim Hartle. It had been five years since Stephen had written his famous paper [14] with Gary Gibbons on quantum field theory in de Sitter space, so Stephen was well-versed in the role that quantum fluctuations can play in an exponentially expanding space. In that paper Gary and Stephen had shown the now-classic result that quantum fluctuations in de Sitter space produce thermal radiation with a temperature where H is the Hubble constant of the de Sitter space, and I use units for whichh = c = k Boltzmann = 1. Stephen was of course also one of the world's experts on cosmological density fluctuations, going back to his seminal work on the evolution of density perturbations in 1966 [25]. I was of course not part of this meeting, but I learned about it from Michael and Paul while I was gathering material for my popular level book * [18], of which the Nuffield Workshop was one of the highlights. Apparently not much in the way of details was discussed at this meeting, but the three physicists discussed a crucially important new idea: the possibility that the structure of the cosmos originated in quantum fluctuations. From the point of view of unity in physics, this is a truly breathtaking idea, proposing that the same quantum phenomena that are central to the study of atoms and subatomic physics are also responsible for the largest structures known to humans. The meeting reportedly ended when Stephen left to shop at F.A.O. Schwarz, the famous toy store. † Later in May, I learned from Paul Steinhardt that he and Michael had come up with a plausible way of estimating the effects of quantum fluctuations, in the context of what was then the universally favored grand unified theory, the minimal SU(5) model. They concluded that the spectrum would be scale-invariant, which is what cosmologists favored [23,56], but that δρ/ρ, the fractional perturbations in the mass density, would be about 10 −16 . The answer that we all wanted was about 10 −4 , so they knew they were not there yet, but they were still trying.
On June 7, 1982, Stephen gave a seminar in Princeton about his density perturbation calculations. I was not there either, but Paul Steinhardt was. The day after the talk, Paul and I had a long telephone conversation in which he gave me a detailed summary of what Stephen had said. I took notes while on the phone, and I still have them. In contrast to Paul's and Michael's result of 10 −16 , Stephen's calculations gave the result of δρ/ρ ∼ 10 −4 , exactly what was wanted. Shortly afterward Stephen circulated a preprint [26] summarizing these calculations. I learned from Stephen that his preprint was actually written before his trip to the U.S. at the beginning of May, although its date is listed as June, when the typing was completed. The abstract of that preprint explained that irregularities would be produced in inflationary models by quantum fluctuations in the scalar field as it ran down the hill of an effective potential diagram. The abstract continued: These would lead to fluctuations in the rate of expansion which would have the right spectrum and amplitude to account for the existence of galaxies and for the isotropy of the microwave background.
So it was a big success! Inflation actually worked, accounting not only for the large-scale uniformity of the Universe, but also for the spectrum of density fluctuations needed to explain the tapestry of cosmic structure.
When Paul and I were discussing Stephen's Princeton seminar on the telephone, Paul relayed to me essentially all the equations that Stephen had presented. These began with equations relating the perturbations during inflation to the quantum fluctuations of the scalar field, as described by the scalar field two-point function. We did not quite understand what Stephen was talking about, but we assumed he was probably right. Stephen also described -in a couple of lines -how density perturbations would evolve from the beginning of inflation up to the present day. These were the sort of calculations that Stephen had done long before; he knew the answers, so he just wrote them down. Again Paul and I assumed that Stephen must have known what he was talking about. There was one step in the calculation, however, where Stephen calculated the time derivative of the scalar field, as it rolled down the hill of the potential energy diagram. That was really the only step in the calculation that I was capable at the time of understanding, but as far as Paul and I could tell, Stephen got it wrong. If we did this calculation our way, the answer became larger by about a factor of 10 4 , giving δρ/ρ about equal to one. So Paul and I were very skeptical of Stephen's result, although, since we did not understand the rest of the calculation, we were not really sure what Stephen was thinking, or whether we were correctly interpreting his equations. From our point of view Stephen's result looked wrong, but we were not absolutely sure.
I then began working on this in detail with So-Young Pi. We closely followed the pattern of Stephen's calculation, trying to make sure that we understood each step. A key element in Stephen's approach was to think of the primary driving force behind the density fluctuations as the fluctuation δt( x) in the time at which inflation ends in different places. The description of density perturbations depends very much on the coordinate system (i.e., the choice of gauge) in which they are described, so the papers on this subject that came out of the Nuffield meeting [27,50,19,3] do not all look the same. While all of these papers got the same answer, the calculations are difficult to compare. The papers by Stephen, Starobinsky, and the one by So-Young and me all focussed on the time delay function δt( x), but they still used somewhat different ways of connecting this function to the final result. The paper by Bardeen, Steinhardt, and Turner used a more complete integration of the full set of equations from perturbative general relativity. I have always found that the time-delay approach, which we learned from Stephen, is the clearest way to understand density fluctuations, as long as one is interested in models with a single scalar field that rolls slowly, so that the Hubble parameter H can be treated as a constant during the period when the perturbations of the relevant wavelengths are generated. The models that interested us in 1982 fell within these restrictions, although the generality of the Bardeen-Steinhardt-Turner approach (and further elaborations -see for example [41]) is needed to analyze the wider range of models that have since become relevant.
By the way, Stephen's paper on density perturbations [27] turns out to be his third most cited paper of all time, so I am personally rather proud to have been vaguely involved in Stephen's third most cited paper! I made a list of Stephen's most cited papers from yesterday's listings on SPIRES (http://www.slac.stanford.edu/spires/hep/), and I was flabbergasted to see that Stephen has 33 papers that have more than a hundred cites; I think this is really astounding. Cites do not mean everything, of course, but when you see this many, you can be sure they mean something!

II. THE 1982 NUFFIELD WORKSHOP
Chronologically, the next event -the big event -was the Nuffield Workshop, which began on Monday, June 21, 1982. I still have my original invitation from the organizers of the meeting, Gary Gibbons and Stephen (see Figure 1). I, of course, didn't care about Gary, but I was so impressed that I received a letter that was actually signed by the famous Stephen Hawking (or at least by someone authorized to use his signature) that I saved it for 20 odd years. The key paragraph in the letter stated the premise of the meeting, which with historical hindsight seems very impressive. It is a really good description of where cosmology was at then, and some of these problems are still problems that we are talking about now: The standard model seems to provide a satisfactory account of the evolution of the universe after 1 sec but it assumes certain initial conditions such as thermal equilibrium, spatial homogeneity and isotropy with small fluctuations, spatial flatness, and the baryon to entropy ratio. The aim of the workshop would be to discuss how these conditions could have arisen from physical processes in the very early universe on the basis of grand unified theories and quantum gravity. Topics covered would include phase transitions, the generation of baryon number, the production of monopoles, primordial black holes and other long lived particles, the existence and nature of the initial singularity, particle creation and the origin of fluctuations.
What I want to talk about today is the success of this conference in making real progress on the issue of the primordial density perturbations.
I still have the transparencies from the talk I gave at the very end of the conference, which summarizes what I knew then about density fluctuations. Figure 2 shows my opening transparency from that talk. I have to confess that the very first thing on the transparency is the potential energy diagram of the minimal SU(5) grand unified theory with a Coleman-Weinberg potential, which at that time we all knew was the correct theory. Its amazing that after 21 years of research we now know so much less -but I guess that is a form of progress, too.
In that same talk, I tried to describe the flow of thoughts on the subject, so I made a transparency to summarize the evolution of our thinking on the values of δρ/ρ as a function of time in 1982 (see Figure 3). We were evaluating δρ/ρ for each mode at "second Hubble crossing," labeled on the chart as 2t = ℓ. Here ℓ is the wavelength in physical (as opposed to comoving) units, and 2t in the radiation-dominated era is equal to H −1 , the Hubble length. (This is the second Hubble crossing, because for each mode there was also a time during inflation when the physical wavelength, which grows exponentially during inflation, crosses the Hubble length, which is approximately constant during inflation.) The scale-invariant Harrison-Zeldovich spectrum is the spectrum for which δρ/ρ at second Hubble crossing has the same value of each mode, and although we were disagreeing on the amplitude, we were all getting approximately scale-invariant spectra.
The chart starts at May 26, 1982, when Paul Steinhart told me that he and Michael Turner were getting δρ/ρ ∼ 10 −16 . The second line refers to Stephen Hawking's seminar in Princeton on June 7, when he announced the result that δρ/ρ ∼ 10 −4 . The following line shows my own answer, δρ/ρ ∼ 50, listed as a private communication from me to me, on June 21. Now I'll admit that showing this transparency may have been a bit egotistical on my part, but after all, these were the days when Stephen and I were both young.
I was working with So-Young Pi, and as I said we started from Hawking's work, struggling to fill in all the details that we needed to get from one line of Stephen's preprint to the next. But we evaluated the time-derivative of the scalar field differently from Stephen, and concluded that in the minimal SU(5) model the perturbations would be far too large, with δρ/ρ ∼ 50 . So-Young and I had started these calculations together, but we didn't have the final answer until I completed the calculations the day after I arrived here in Cambridge. Cambridge is a great place to do calculations. They gave us rooms in Sidney Sussex College with thick walls and thick doors and no telephones, so I worked late into the night and finished the calculation the next morning, after breakfast, behind those thick doors. I have always thought Cambridge is a fantastic place to do physics. Anyway I got the answer shown in the third line of the chart, δρ/ρ ∼ 50, which means that the model did not work. It is interesting to wonder whether I was disappointed with my result, since inflation was my baby.
The answer is that, since inflation had not really caught on yet in 1982, I was absolutely thrilled to find an answer that I was pretty sure corrected a calculation of Stephen's, which to me was more of a victory than if inflation had worked! In the end I have had some of the benefits of both options, since the minimal SU(5) grand unified theory is now ruled out by the absence of observed proton decays, and we can construct other particle theory models that allow the density perturbation amplitude to turn out right.
Continuing, the next line shows the first seminar on density fluctuations at the Nuffield Workshop, by Alexei Starobinski, on Monday June 28, at the beginning of the second week. My chart shows him at δρ/ρ ∼ 10 −2 , but he later clarified that he found that the perturbations were of order unity, and hence the real conclusion is simply that perturbation theory could not be trusted.
Then Stephen gave his seminar, on Tuesday June 29, a little more than a week into the conference. Before this time I had had one conversation with Stephen about density perturbations, but that conversation was cut off by the start of the next seminar; I remember that this was my seminar so I did not think I should miss it! Stephen was continuing to argue that his original preprint was correct, but Paul Steinhardt and I were having trouble understanding Stephen's line of reasoning. Stephen's seminar was impishly titled "The End of Inflation," which of course could have meant that he was going to discuss how the inflationary era would end, or it could have meant that he was going to argue that the inflationary theory is dead. Stephen always liked to generate surprises. When Stephen began his seminar, Paul and I were prepared to pounce at the point where we thought he had made a mistake, hoping that the ensuing discussion could settle the issue. But when Stephen reached this point in the seminar, very near the end, he jolted us by not using the argument we expected, but instead substituted a new calculation which in fact agreed with our calculation for the time derivative of the scalar field. He then agreed with our answer, that the density perturbations were indeed far too large. He expressed his answer as α/C, where α is the fine-structure constant of the grand unified theory, and C was a dimensionless constant that was not calculated, which he referred to in his talk as a "fudge factor." In his talk, Stephen concluded that we would need C ∼ 100, or else the inflationary scenario would have to be abandoned. In the published version of Stephen's paper [27], the abstract began identically but ended differently: These [irregularities in the scalar field] would lead to fluctuations in the rate of expansion which would have the right spectrum to account for the existence of galaxies. However the amplitude would be too high to be consistent with observations of the isotropy of the microwave background unless the effective coupling constant of the Higgs scalar was very small.
'Very small', as we now know, means about 10 −13 or something close to that. I have never known when it was that Stephen changed his mind, or whether he circulated this preprint just to tease us from the beginning. He certainly did not change his mind while he was giving his talk, so he must have decided earlier that his original preprint had the wrong answer. But he did not tell any of us. Stephen recently told me that he initially calculated the time derivative of the scalar field when it was halfway down the hill, but talking to people at the workshop made him realize that it should be evaluated at first Hubble crossing, which gives much larger perturbations. When Stephen gave the talk, he proceeded without a pause, with no suggestion that the result he was presenting was in contradiction of his own preprint.
As my chart shows, by the time of my talk Bardeen, Steinhardt, and Turner had invented a more accurate way to solve their equations, and found an answer in agreement with the rest of us. The result that we all found was roughly where δφ ∼ H is the quantum fluctuation in φ, C is a constant of order unity which depends on detailed normalization conventions, andφ cl is the proper time-derivative of the function that describes the classical evolution of the inflaton field. The right-hand side is evaluated at first Hubble crossing, the instant during the inflationary era when the physical wavelength of the mode under consideration is equal to H −1 .

III. OBSERVATIONAL EVIDENCE FOR INFLATION
At the time of the Nuffield meeting, I thought it was great fun and excitement to try to calculate the predictions for the density perturbations resulting from inflation, but I never believed that anyone would actually measure these density perturbations. I guess I am speaking mainly for myself, but I suspect that the others felt the same way. I knew that we had some idea of what density perturbations were needed to explain galaxy formation, and I'm sure I believed that such estimates would improve with time. But I never thought we would have direct measurements of the density perturbations coming from the microwave background. The radiation itself is incredibly weak; it is, we should keep in mind, thermal radiation at 3 K, which is about 100,000,000 times weaker than thermal radiation at room temperature, which is itself pretty weak. And the nonuniformities that we are talking about are at the level of only about one part in 100,000. So I found it absolutely astounding when the COBE results were published in 1992, and I was even more astounded by the recent results of Boomerang, Maxima, DASI, and CBI.
Today the measurements of the cosmic microwave background (CMB) and the perturbations that they exhibit are truly extraordinary. Figure 4 is a graph of the most recent data that was released by the Boomerang group [42], the Maxima group [32], the DASI group [22], and the CBI group [43]. These experiments measure the nonuniformities ∆T (θ, φ) of the CMB radiation, which are described by their expansion in spherical harmonics: For each ℓ one defines and then The graphs show the measurements of the C ℓ 's in microkelvin 2 , as a function of multipole ℓ. The graph also shows a theoretical prediction, corresponding to the "Joint model" of Ref. [43], which was computed using the computer code of Seljak and Zaldarriaga, CMB-FAST [47]. The curves show a series of well-defined "acoustic" peaks, caused by the oscillations of the cosmic fluid of photons, neutrinos, electrons, protons, and dark matter. Roughly speaking the image of the cosmic background radiation on the sky is a snapshot of what the universe looked like at the time of last scattering, a time about 400,000 years after the big bang, at which the plasma of the early universe neutralized to form a transparent gas. Although the fluctuations of all wavelengths are imprinted on the CMB at the same time, it is nonetheless a conceptually useful approximation to think of the graph of C ℓ 's versus ℓ as if it were a graph of the fluctuations of the early universe as a function of time. The shorter wavelength (i.e., higher ℓ) fluctuations evolve faster, so at the time of last scattering they have undergone more oscillations and are therefore in a later stage of their evolution than their longer wavelength cousins. The lower amplitude at higher ℓ is partly due to the damping of these oscillations, and partly due to the fact that the time of last scattering is not unique, since the transition from opacity to transparency is actually somewhat gradual. Thus the CMB image has some similarities to a photographic time exposure, and the small scale details are therefore blurred. The probability distribution for the time of last scattering is shown in Figure 5, calculated ‡ for exactly the same parameters as the curves in Figure 4. The median time of last scattering is 388,000 years, the peak of the curve is at 367,000 years, and the full width at ‡ The visibility function shown here was calculated using CMBFAST, but the program had to be modified to print out this information, which normally is used internally. I also modified the code slightly to keep track of proper time, as opposed to conformal time. The ionization evolution was calculated using the RECFAST subroutine distributed with CMBFAST, which was written by D. Scott, based on calculations by Seager, Sasselov, and Scott [46].
half maximum is 113,000 years. (The mean was computed as 475,000 years, but the calculation was influenced significantly by a long late-time tail of the probability distribution, extending to billions of years; it is not clear if this calculation is reliable or relevant.) The theoretical curve shown in Figure 4 has an amplitude which is normalized to the data; in practice inflationary models do not make any prediction for the overall amplitude of the fluctuations, though we could in principle make a prediction if we really knew the potential energy function of the inflaton field, the scalar field that drives inflation.
However, everything else about this curve is pretty well fixed either by inflation, or by astronomical determinations of cosmological parameters. For example, the curves depend sensitively on Ω tot = ρ tot /ρ c , where ρ tot is the total mass density, and the critical mass density ρ c is defined to be that density which corresponds to a flat spatial geometry. § ρ c is related to the Hubble constant H by ρ c = 3H 2 /(8πG), where G is Newton's gravitational constant. Inflation predicts that Ω tot = 1, and the curves are drawn for Ω tot = 1. If Ω tot were decreased from 1, all the peaks would shift to the right. If Ω tot were 0.3, as was widely believed five years ago, the first peak would be at about ℓ = 400.
The curve also depends on the present value of the Hubble constant, which was taken to be H ≡ 100 h·km·sec −1 ·Mpc −1 , with h = 0.68. This value is completely consistent with the Hubble Key Project value [13], h = 0.72 ± 0.08. An increase in h would push all the peaks downward and toward the left.
The density of baryons in the universe also affects the predicted curves. This density is usually quantified by the product Ω b h 2 . (Note that Ω b h 2 ≡ ρ b h 2 /ρ c , so the explicit factor of h 2 cancels the h-dependence of ρ c , and consequently Ω b h 2 is really just an indirect way of describing ρ b .) A high density of baryons suppresses the second peak, and indeed when the first Boomerang results [4] were released in 2000, the absence of an apparent second peak led some to speculate (see, for example, Ref. [31]) that the universe might contain 50% more baryons than expected on the basis of big bang nucleosynthesis. This problem soon dissolved with the appearance of new data, and the fit shown in Figure 4 uses Ω b h 2 = 0.020, in perfect agreement with nucleosynthesis-based estimate of Burles, Nollett, and Turner [11], Ω b h 2 = 0.020 ± 0.002 (95% confidence).
The scalar power law index n s is predicted to be very near one for simple inflationary models, corresponding to a scale-invariant Harrison-Zeldovich [23,56] spectrum; the data is fit with n s = 0.95. τ c = 0 means that no re-ionization of the intergalactic medium is assumed, and the values used for the densities of cold dark matter (Ω CDM = 0.257) and "dark energy" (Ω Λ = 0.7) are consistent with the supernova observations [45,44].
So, except for the height of the curve, the data shown in Figure 4 can be essentially predicted on the basis of other measurements and on the inflationary model. I consider this a spectacular success. I am amazed that this data can be measured, and that it agrees so well with what we predicted back in 1982. It seems to show not only that inflation is correct, but that the simplest form of slow-roll inflation is correct. Given the amount of flexibility § Until recently it was common to say that the critical density was that density which put the universe just on the borderline between eternal expansion and eventual collapse. This definition was never generally accepted as a technical definition, but was nonetheless often used in lectures, especially those intended for a non-technical audience. If the only materials present are normal matter, dark matter, and radiation, then the two definitions are equivalent. However, since we now believe that the universe contains a large amount of "dark energy" with negative pressure, these definitions are no longer equivalent. We must therefore keep in mind that the critical density is defined in terms of the spatial geometry of the universe.
that inflationary models allow, and also the uncertainties that exist in the data, I suspect that probably it is fortuitous that the agreement is as good as what we see. My guess is that we are going to see some disagreements before things really fit together tightly, but nonetheless we are currently seeing a very spectacular agreement which strongly suggests that we are on the right course.

IV. ETERNAL INFLATION
For the remaining half of my talk, I want to discuss an issue related to the eternal nature of inflation. This may be only vaguely related to the topic that I was asked to talk about, inflation and cosmological perturbations, but there is a connection. The density perturbations arising from inflation are never truly scale invariant, but typically grow slowly at large wavelengths. Eternal inflation is really a consequence of the very long wavelength, high amplitude tail of the density perturbation spectrum of inflation. At extremely long wavelengths these perturbations are usually so large that they prevent inflation from ending at all in some places, leading to what is called eternal inflation.
My main goal is to describe a singularity theorem that I recently proved in collaboration with Alex Vilenkin and Arvind Borde [5]. But first, I would like to give a brief explanation of how eternal inflation works.
There are basically two versions of inflation, and consequently two answers to the question of why inflation is eternal (see Figure 6). In the case of new inflation, the exponential expansion occurs as the scalar field rolls from the peak of a potential energy diagram down to a trough. (I refer to the state for which the scalar field is at the top of the hill as a false vacuum, although this is a slight change from the original definition of the phrase.) The eternal aspect occurs while the scalar field is hovering around the peak. The first model of this type was constructed by Steinhardt [52] in 1983, and later that year Vilenkin [54] showed that new inflationary models are generically eternal. The key point is that, even though classically the field would roll off the hill, quantum-mechanically there is always an amplitude, a tail of the wave function, for it to remain at the top. If you ask how fast does this tail of the wave function fall off with time, the answer in almost any model is that it falls off exponentially with time, just like the decay of most metastable states [20]. The time scale for the decay of the false vacuum is controlled by the negative mass-squared of the scalar field when it is at the top of the hill in the potential diagram. This is an adjustable parameter as far as our use of the model is concerned, but m has to be small compared to the Hubble constant or else the model does not lead to enough inflation. So, for parameters that are chosen to make the inflationary model work, the exponential decay of the false vacuum is slower than the exponential expansion. Even though the false vacuum is decaying, the expansion outruns the decay and the total volume of false vacuum actually increases with time rather than decreases. Thus inflation does not end everywhere at once, but instead inflation ends in localized patches, in a succession that continues ad infinitum. Each patch is essentially a whole universe -at least its residents will consider it a whole universe -and so inflation can be said to produce not just one universe, but an infinite number of universes. These universes are sometimes called bubble universes, but I prefer to use the phrase "pocket universe," to avoid the implication that they are approximately round. (While bubbles formed in first-order phase transitions are round [12], the local universes formed in eternal new inflation are generally very irregular, as can be seen for example in the two-dimensional simulation by Vanchurin, Vilenkin, and Winitzki in Fig. 2 of Ref. [53].)

FIG. 6: Eternal inflation from both a new inflationary potential (left) and a chaotic inflationary potential (right).
In the context of chaotic inflationary models, as developed by Andrei Linde (who both proposed the models [35,36] and showed that they are eternal [37,38,16,39]), the situation is slightly more complicated. Inflation is occurring as the scalar field rolls down a hill of the potential energy diagram, starting high on the hill. As the field rolls down the hill, quantum fluctuations will be superimposed on top of the classical motion. The best way to think about this is to ask what happens during one time interval of duration ∆t = H −1 (one Hubble time), in a region of one Hubble volume H −3 . Suppose that φ 0 is the average value of φ in this region, at the start of the time interval. By the definition of a Hubble time, we know how much expansion is going to occur during the time interval: exactly a factor of e. (This is the only exact number in today's talk, so I wanted to emphasize the point.) That means the volume will expand by a factor of e 3 . One of the deep truths that one learns by working on inflation is that e 3 is about equal to 20, so the volume will expand by a factor of 20. Since correlations typically extend over about a Hubble length, by the end of one Hubble time, the initial Hubble-sized region grows and breaks up into 20 independent Hubble-sized regions.
As the scalar field is classically rolling down the hill, the classical change in the field ∆φ cl during the time interval ∆t is going to be modified by quantum fluctuations ∆φ qu , which can drive the field upward or downward relative to the classical trajectory. For any one of the 20 regions at the end of the time interval, we can describe the change in φ during the interval by ∆φ = ∆φ cl + ∆φ qu .
In lowest order perturbation theory the fluctuation is treated as a free quantum field, which implies that ∆φ qu , the quantum fluctuation averaged over one of the 20 Hubble volumes at the end, will have a Gaussian probability distribution, with a width of order H/2π [55,34,50,51]. There is then always some probability that the sum of the two terms on the right-hand side will be positive -that the scalar field will fluctuate up and not down. As long as that probability is bigger than 1 in 20, then the number of inflating regions with φ ≥ φ 0 will be larger at the end of the time interval ∆t than it was at the beginning. This process will then go on forever, so inflation will never end. Thus, the criterion for eternal inflation is that the probability for the scalar field to go up must be bigger than 1/e 3 ≈ 1/20. For a Gaussian probability distribution, this condition will be met provided that the standard deviation for ∆φ qu is bigger than 0.61|∆φ cl |. Using ∆φ cl ≈φ cl H −1 , the criterion becomes Comparing with Eq. (2), we see that the condition for eternal inflation is equivalent to the condition that δρ/ρ on ultra-long length scales is bigger than a number of order unity. The probability that ∆φ is positive tends to increase as one considers larger and larger values of φ, so sooner or later one reaches the point at which inflation becomes eternal. If one takes, for example, a scalar field with a potential then the de Sitter space equation of motion in flat Robertson-Walker coordinates (ds 2 = −dt 2 + e 2Ht d x 2 ) takes the formφ where spatial derivatives have been neglected. In the "slow-roll" approximation one also neglects theφ term, soφ ≈ −λφ 3 /(3H), where the Hubble constant H is related to the energy density by where M p ≡ 1/ √ G is the Planck mass. Putting these relations together, one finds that the criterion for eternal inflation, Eq. (8), becomes Since λ must be taken very small, on the order of 10 −12 , for the density perturbations to have the right magnitude, this value for the field is generally well above the Planck scale. The corresponding energy density, however, is given by which is actually far below the Planck scale. So for these reasons we think inflation is almost always eternal. I think the inevitability of eternal inflation in the context of new inflation is really unassailable -I do not see how you could possibly avoid it, assuming that the rolling of the scalar field off the top of the hill is slow enough to allow inflation to be successful. The argument in the case of chaotic inflation is a bit more approximate, and some people have questioned it, but I still believe it has to work because the criterion that has to be satisfied is so mild. For eternal inflation to set in, all one needs is that the probability for the field to increase in a given Hubble-sized volume during a Hubble time interval is larger than 1/20.

V. A NEW SINGULARITY THEOREM
Eternal inflation implies that once inflation starts, it never stops. This leads to the question: can inflation by itself be the complete theory of cosmic origins? Can inflation be eternal into the past as well as the future, allowing a model which on very large scales is steady state, eliminating the need for a beginning? The answer I believe is no, although I would not claim that we have a rock-solid proof. Borde, Vilenkin, and I [5] have proven a rigorous theorem, which I will describe, and this theorem certainly shows that the simplest type of inflationary models still require a beginning, even though they are eternal into the future. The difficulty is that we have no way of discussing the class of all possible inflationary models, so we cannot say that our theorem applies to all cases. Our singularity theorem was certainly inspired by Stephen's famous work on singularity theorems, which established the value of such theorems, so it is very fitting that I describe this theorem at a symposium in honor of Stephen's 60th birthday. And I am sure that our conclusions fall on the side that Stephen would prefer, since Stephen has put much effort into studying the quantum origin of the universe, a subject which could have been bypassed if inflationary models could avoid a beginning.
I will not try to state the theorem immediately, since it will be easier later, after some definitions have been put forward.
An unusual feature of the new singularity theorem is that it avoids any mention of things such as energy conditions, which are crucial assumptions for other formulations of singularity theorems. In particular, in the context of eternal inflation, the key papers by Borde and Vilenkin [7,6,8,9] proved several different theorems, all of which invoked the weak energy condition. This is the condition that n µ n ν T µν ≥ 0, where T µν is the energy-momentum tensor and n µ is any timelike vector. For a perfect fluid, the condition is equivalent to assuming that ρ ≥ 0 and ρ + p ≥ 0, where ρ is the energy density and p the pressure. The weak energy condition is always valid classically, but it is nevertheless violated by quantum fluctuations. In particular, in a perfect de Sitter space the quantity ρ + p is identically equal to zero. In a quantum description, however, the vacuum is never an eigenstate of ρ + p, so the quantity must fluctuate, but it must still average to its classical value of zero. It therefore fluctuates both positively and negatively, thereby violating the weak energy condition about half the time [10,21]. The nice thing about the new theorem is that it is purely kinematical; it really depends in no way on the dynamics of general relativity, but only on the redshifting of velocities in an expanding universe. It seems rather amazing, however, that we can learn something useful by considering only relativistic kinematics. Crudely speaking, the theorem says that if the universe expands fast enough, then it cannot possibly be geodesically complete to the past. In a few minutes I will be able to define what it means for the universe to expand fast enough.
We wish to prove a theorem that will apply to any universe, no matter how inhomogeneous or anisotropic, so we want to think of the expansion as a local phenomenon. To describe this expansion, we need to adopt a local definition of the Hubble parameter. One way to define a local Hubble parameter would be to imagine measuring the velocities of particles moving with the Hubble flow within some small neighborhood, and then one could define a local Hubble parameter in terms of the divergence of the velocity field. For the purpose of our theorem, however, we have found it useful to consider an even more local definition of the Hubble parameter, one that can be measured by a single geodesic observer traveling through the universe. Our hypothetical observer would never get a job at the Keck because he does not know how to use a telescope. He is completely myopic, able to measure the velocity only of those particles that intersect his own trajectory. If he is at rest relative to the local Hubble expansion he will not be able to measure anything, but if he is moving relative to the Hubble expansion he will pass a succession of particles, and will infer a Hubble expansion parameter from the rate at which those particles are separating from each other. We will call the particles that the observer passes "comoving test particles," but for the purpose of the theorem it is not really necessary that these particles exist. All that is necessary is that the worldlines of these hypothetical particles can be defined on the background spacetime, and that each worldline has zero acceleration at the instant that it intersects the observer's trajectory. The observer will be assumed to be traveling on a geodesic, either timelike or null. As illustrated in Figure 7, the four-velocity of the observer will be called v µ (τ ), where τ will denote proper time for the case of a timelike observer, and an affine parameterization for the case of a null observer. In either case v µ (τ ) ≡ dx µ /dτ . The comoving test particle passed by the observer at time τ will be moving at a four-velocity called u µ (τ ). To define the Hubble parameter that the observer measures at time τ , the observer focuses on two particles, one that he passes at time τ , and one at τ + ∆τ , where in the end he takes the limit ∆τ → 0. The Hubble parameter is defined by where ∆v radial is the radial component of the relative velocity between the two particles, and ∆r is their distance, where both quantities are computed in the rest frame of one of the test particles, not in the rest frame of the observer. Note that this definition reduces to the usual one if it is applied to a homogeneous isotropic universe, but it can be defined for any spacetime in which one has identified a geodesic observer and a family of comoving test particle trajectories. We will also be interested in the relative velocity between the test particles and the observer, which can be measured by For the case of timelike observers, γ corresponds to the usual Lorentz factor of special relativity Now the key point is that the observer, if H is positive, will see himself redshifting in an expanding universe, which means that his velocity relative to the test particles will be slowing down. Looking into the past, however, the velocity will become blueshifted, becoming faster and faster as one follows the trajectory further into the past. We will find that under many circumstances this velocity reaches the limiting speed of light in a finite proper time, and then the trajectory can be continued no further.
We are accustomed to calculating the redshifting of particle velocities by writing the metric for the background spacetime and then solving the geodesic equations. However, the slowdown is a kinematical effect that is much simpler than one might infer from seeing the standard calculation. To understand the logic, it is useful to begin with a simple onedimensional nonrelativistic analogy. Consider a stream of cars traveling on a straight road, and imagine that they are moving apart from each other, so the cars are the comoving test particles of an expanding universe of cars. You will play the role of the geodesic observer in this thought experiment, and to simplify the description we can imagine that the rest frame of the road coincides with your rest frame; i.e., you are standing at rest just alongside the road. Suppose that one car goes by and you measure its speed relative to you as 40 miles per hour. When the next car comes by, you know that its speed is not going to be 50 mph, because that would mean it was catching up to the first car, and we already said that the cars were getting further apart. If they are getting further apart, then the second car has to be moving slower than the first, and the rate that they are getting further apart is directly proportional to the difference between the speeds. So the Hubble expansion rate that you would measure for the stream of cars is directly proportional to the rate at which the relative speed between you and the cars is decreasing.
When one increases the number of space dimensions from one to three, one might expect that the simple relationship would break down. Now the velocity of the comoving test particles relative to the observer can change both in magnitude and direction, so the result is less obvious. However, if one remembers that the Hubble parameter is defined in terms of the radial component of the relative velocity, a short calculation shows that the onedimensional result is still valid: the Hubble parameter is directly proportional to the rate of change of the relative speed between the observer and the comoving test particles, with changes in direction giving no contribution. Finally, one wants to generalize the calculation to allow for relativistic velocities. As Borde, Vilenkin, and I showed in our paper, one finds again that the Hubble parameter as defined by Eq. (14) can be related directly to the rate of change of the relative velocities. Specifically, in our paper we defined a quantity which I like to call the slowness parameter. Note that for the timelike case, a relative velocity of zero (γ = 1) corresponds to infinite slowness (F (γ) = ∞), and that in both cases the limiting relative velocity γ → ∞ corresponds to zero slowness (F (γ) = 0). The existence of a maximum possible relative velocity corresponds to a minimum possible slowness. The definition (17) has been chosen so that the minimum value is zero. In our paper we showed that the Hubble parameter is related to the slowness by the simple relation, That is, the Hubble parameter is equal to the rate of change of the slowness. If H > 0 the universe expands, and the slowness increases, which means that the velocity of the geodesic observer relative to the comoving test particles is redshifted. If one looks backwards along the geodesic, however, one sees a blueshift. The velocity of the geodesic observer relative to the comoving test particles increases as one looks backwards, and the slowness decreases. However, the slowness cannot fall below zero. Once the slowness reaches zero it cannot get any lower, so according to Eq. (18) it is impossible to continue the geodesic any further if H continues to be positive. The verbal argument of the previous paragraph can be translated into a rigorous inequality by integrating Eq. (18) over proper time, and using the fact that the value of F (γ) at the lower limit of integration is always nonnegative. For timelike geodesics, this leads to the result where γ refers to the value of u µ v µ at the final time τ f , and the inequality holds for any value of the lower limit of integration. Thus, the integral of H along a backwards-going geodesic is limited by an expression that depends only on the final value of the relative velocity between the geodesic observer and the comoving test particles. The limit disappears only for the case in which the geodesic observer is at rest (relative to the comoving test particles) at the final time, in which case the right-hand-side of Eq. (19) is infinite. The right-hand-side of Eq. (19) can be algebraically rewritten as where the first term dominates for low velocities, and the second term dominates for large velocities.
For null observers the result looks even simpler, but it depends on the normalization chosen for the affine parameter. If we normalize it by the convention that γ = dt/dτ = 1 at the final time τ f , where t is the time measured by comoving observers, then the bound is simply Eqs. (19) and (21) are technically our final results, but I would like to say a few words about what we think these results mean.
One illustration of the theorem is its application to de Sitter space, especially as described in flat Robertson-Walker coordinates: whereH is a constant. Using the worldlines x = constant as comoving test particle trajectories, this spacetime would give H =H at all points in the coordinate system, independent of the geodesic observer worldine used to define the measurement. As is well-known, however, at least to readers of Hawking and Ellis [28], these coordinates cover only half of de Sitter space. The locus t = −∞ describes a null hypersurface that forms the boundary of the two halves. The comoving geodesics ( x = constant) have infinite length within the half of de Sitter space described by these coordinates, but our theorem implies (as can be verified by direct calculation) that any other backwards-going timelike geodesic reaches t = −∞ in a finite amount of proper time. In fact, any noncomoving backwards-going timelike geodesic reaches the t = −∞ hypersurface with γ = ∞, and therefore just saturates the bound implied by our theorem. If we follow such a backwards-going geodesic, it is of course possible to continue it into the other half of the de Sitter space, since de Sitter space as a whole is geodesically complete. Such an extended geodesic would have infinite proper length. Furthermore, since de Sitter space is homogeneous, at any location along the infinite backwards-going geodesic it would be possible to construct a neighborhood containing comoving test particle trajectories that would make the universe appear to be expanding, with H =H. Our theorem guarantees, however, that it is not possible to define such comoving test particle trajectories globally, so that H =H everywhere along the backwards-going geodesic. In fact, since the bound is saturated as the trajectory reaches the t = −∞ hypersurface, the theorem guarantees that H must turn negative beyond this hypersurface, no matter how the comoving test particle trajectories might be chosen. Thus, comoving test particle trajectories can be chosen so that H =H anywhere in the de Sitter space, but this condition cannot be enforced everywhere at once. One simple description of the entire de Sitter space is the closed universe description, ds 2 = −dt 2 + cosh 2 (Ht) dr 2 1 − r 2 + r 2 dθ 2 + sin 2 θ dφ 2 .
For the comoving test particle trajectories (r, θ, φ) = constant, H =H tanh(Ht), and thus the space is contracting at early times and expanding only at late times. This is geodesically complete, but does not constitute a model of inflation that is eternal into the past, since in the past the model is contracting and not inflating. If the false vacuum that supports the de Sitter space is only metastable, it would decay completely during the infinite period of contraction, so inflation would never take place.
To describe the implications of our theorem succinctly, it is useful to define a concept that we call uniformly bounded expansion. A region of spacetime is said to be undergoing uniformly bounded expansion if it is possible to define a congruence of comoving test particle trajectories with the property H > H min everywhere in the region, for some H min > 0. The half of de Sitter space described by Eq. (22) provides an example of a spacetime with uniformly bounded expansion throughout. The bounds of Eqs. (19) and (21) imply that any spacetime that contains at least one noncomoving past-directed geodesic exhibiting uniformly bounded expansion cannot be geodesically complete in the past.

VI. THE ORIGIN OF THE UNIVERSE
Now what does this say about the universe? First let me point out a few disclaimers, to make it clear what the theorem does not say. It does not imply that an eternally inflating model must have a unique beginning, and it also does not imply that there is an upper bound on the length of all backward-going geodesics from a given point. Our theorem places no bound on the length of the comoving trajectories, and we know from the de Sitter space example of Eq. (22) that such trajectories really can be infinite. Furthermore, our theorem allows for the possibility of models that have regions of contraction interspersed among regions of expansion, so that Eqs. (19) and (21) can perhaps be satisfied for infinitely long backwards-going geodesics. I should also mention here for completeness that Aguirre and Gratton [1] responded to our paper by suggesting a geodesically complete model which uses the full de Sitter space, but proposes that the thermodynamic arrow of time is reversed in the two halves, so that both halves appear to be expanding. This successfully evades our theorem, but for my taste it seems like an extravagant assumption.
In summary, the theorem by Borde, Vilenkin, and me does show that any inflating model that is globally expanding must be geodesically incomplete in the past. This theorem certainly applies to the simplest models of eternal inflation, in which an approximately de Sitter region grows exponentially, with pieces of it breaking off and decaying to form pocket universes. Geodesic incompleteness of the inflating region implies that there must be some other physics introduced at the beginning, to explain what happens at the past boundary of the inflating region. The most likely possibility, from my point of view, is some kind of quantum origin, which of course touches on Stephen's work. This means that even in eternally inflating models, a beginning is necessary. I would expect that the details of this beginning would be washed out by the inflationary evolution, but nonetheless the model still requires a beginning of some sort, something like the Hartle-Hawking [24] or Hawking-Turok [30] wavefunction of the universe.