Abstract
The basic workings of inflationary models are summarized, along with the arguments that strongly suggest that our universe is the product of inflation. I describe the quantum origin of density perturbations, giving a heuristic derivation of the scale invariance of the spectrum and the leading corrections to scale invariance. The mechanisms that lead to eternal inflation in both new and chaotic models are described. Although the infinity of pocket universes produced by eternal inflation are unobservable, it is argued that eternal inflation has real consequences in terms of the way that predictions are extracted from theoretical models. Although inflation is generically eternal into the future, it is not eternal into the past: it can be proven under reasonable assumptions that the inflating region must be incomplete in past directions, so some physics other than inflation is needed to describe the past boundary of the inflating region. The ambiguities in defining probabilities in eternally inflating spacetimes are reviewed, with emphasis on the youngness paradox that results from a synchronous gauge regularization technique.
A. H. Guth
]ALAN H. GUTH
Massachusetts Institute of Technology
Chapter \thechapter Inflation
1 Introduction
I will begin by summarizing the basics of inflation, including a discussion of how inflation works, and why many of us believe that our universe almost certainly evolved through some form of inflation. This material is mostly not new, although the observational evidence in support of inflation has recently become much stronger. Since observations of the cosmic microwave background (CMB) power spectrum have become so important, I will elaborate a bit on how it is determined by inflationary models. Then I will move on to discuss eternal inflation, showing how once inflation starts, it generically continues forever, creating an infinite number of “pocket” universes. If inflation is eternal into the future, it is natural to ask if it can also be eternal into the past. I will describe a theorem by Borde, Vilenkin, and me (Borde, Guth, & Vilenkin 2003) which shows under mild assumptions that inflation cannot be eternal into the past, and thus some new physics will be necessary to explain the ultimate origin of the universe.
2 How Does Inflation Work?
The key property of the laws of physics that makes inflation possible is the existence of states with negative pressure. The effects of negative pressure can be seen clearly in the Friedmann equations,
(\thechapter.1a)  
(\thechapter.1b)  
and 

(\thechapter.1c) 
where
(2) 
Here is the energy density, is the pressure, is Newton’s constant, an overdot denotes a derivative with respect to the time , and throughout this paper I will use units for which . The metric is given by the RobertsonWalker form,
(3) 
where is a constant which by rescaling can always be taken to be 0 or 1.
Eq. (1) clearly shows that a positive pressure contributes to the decceleration of the universe, but a negative pressure can cause acceleration. Thus, a negative pressure produces a repulsive form of gravity.
Furthermore, the physics of scalar fields makes it easy to construct states of negative pressure, since the energymomentum tensor of a scalar field is given by
(4) 
where is the metric, with signature , and is the potential energy density. The energy density and pressure are then given by
(5)  
(6) 
Thus, any state which is dominated by the potential energy of a scalar field will have negative pressure.
Alternatively, one can show that any state which has an energy density that cannot be easily lowered must have a negative pressure. Consider, for example, a state for which the energy density is approximately equal to a constant value . Then, if a region filled with this state of matter expanded by an amount , its energy would have to increase by
(7) 
This energy must be supplied by whatever force is causing the expansion, which means that the force must be pulling against a negative pressure. The work done by the force is given by
(8) 
where is the pressure inside the expanding region. Equating the work with the change in energy, one finds
(9) 
which is exactly what Eqs. (5) and (6) imply for states in which the energy density is dominated by the potential energy of a scalar field. (One can derive the same result from Eq. (3), by considering the case for which .)
In most inflationary models the energy density is approximately constant, leading to exponential expansion of the scale factor. By inserting Eq. (9) into (1), one obtains a second order equation for for which the latetime asymptotic behavior is given by
(10) 
In the original version of the inflationary theory (Guth 1981), the state which drove the inflation involved a scalar field in a local (but not global) minimum of its potential energy function. A similar proposal was advanced slightly earlier by Starobinsky (1979, 1980) as an (unsuccessful) attempt to solve the initial singularity problem, using curved space quantum field theory corrections to the energymomentum tensor to generate the negative pressure. The scalar field state employed in the original version of inflation is called a false vacuum, since the state temporarily acts as if it were the state of lowest possible energy density. Classically this state would be completely stable, because there would be no energy available to allow the scalar field to cross the potential energy barrier that separates it from states of lower energy. Quantum mechanically, however, the state would decay by tunneling (Coleman 1977; Callan & Coleman 1977; Coleman & De Luccia 1980). Initially it was hoped that this tunneling process could successfully end inflation, but it was soon found that the randomness of the bubble formation when the false vacuum decayed would produce disastrously large inhomogeneities. Early work on this problem by Guth and Weinberg was summarized in Guth (1981), and described more fully in Guth & Weinberg (1983). Hawking, Moss, & Stewart (1982) reached similar conclusions from a different point of view.
This “graceful exit” problem was solved by the invention of the new inflationary universe model by Linde (1982a) and by Albrecht & Steinhardt (1982). New inflation achieved all the successes that had been hoped for in the context of the original version. In this theory inflation is driven by a scalar field perched on a plateau of the potential energy diagram, as shown in Fig. 1. Such a scalar field is generically called the inflaton. If the plateau is flat enough, such a state can be stable enough for successful inflation. Soon afterwards Linde (1983a, 1983b) showed that the inflaton potential need not have either a local minimum or a gentle plateau: in the scenario he dubbed chaotic inflation, the inflaton potential can be as simple as
(11) 
provided that begins at a large enough value so that inflation can occur as it relaxes. A graph of this potential energy function is shown as Fig. 2. The evolution of the scalar field in a RobertsonWalker universe is described by the general relativistic version of the KleinGordon equation,
(12) 
For late times the term becomes negligible, and the evolution of the scalar field at any point in space is similar to the motion of a point mass evolving in the potential in the presence of a damping force described by the term.
For simplicity of language, I will stretch the meaning of the phrase “false vacuum” to include all of these cases; that is, I will use the phrase to denote any state with a large negative pressure.
Many versions of inflation have been proposed. In particular, versions of inflation that make use of two scalar fields (i.e., hybrid inflation (Linde 1991, 1994; Liddle & Lyth 1993; Copeland et al. 1994; Stewart 1995) and supernatural inflation (Randall, Soljačić, & Guth 1996)) appear to be quite plausible. Nonetheless, in this article I will discuss only the basic scenarios of new and chaotic inflation, which are sufficient to illustrate the physical effects that I want to discuss.
The basic inflationary scenario begins by assuming that at least some patch of the early universe was in this peculiar false vacuum state. To begin inflation, the patch must be approximately homogeneous on the scale of , as defined by Eq. (10). In the original papers (Guth 1981; Linde 1982a; Albrecht & Steinhardt 1982) this initial condition was motivated by the fact that, in many quantum field theories, the false vacuum resulted naturally from the supercooling of an initially hot state in thermal equilibrium. It was soon found, however, that quantum fluctuations in the rolling inflaton field give rise to density perturbations in the universe, and that these density perturbations would be much larger than observed unless the inflaton field is very weakly coupled (Starobinsky 1982; Guth & Pi 1982; Hawking 1982; Bardeen, Steinhardt, & Turner 1983). For such weak coupling there would be no time for an initially nonthermal state to reach thermal equilibrium. Nonetheless, since thermal equilibrium describes a probability distribution in which all states of a given energy are weighted equally, the fact that thermal equilibrium leads to a false vacuum implies that there are many ways of reaching a false vacuum. Thus, even in the absence of thermal equilibrium—even if the universe started in a highly chaotic initial state—it seems reasonable to assume that some small patches of the early universe settled into the false vacuum state, as was suggested for example by Guth (1982). Linde (1983b) pointed out that even highly improbable initial patches could be important if they inflated, since the exponential expansion could still cause such patches to dominate the volume of the universe. If inflation is eternal, as I will discuss in Sec. 5, then the inflating volume increases without limit, and will presumably dominate the universe as long as the probability of inflation starting is not exactly zero.
Once a region of false vacuum materializes, the physics of the subsequent evolution is rather straightforward. The gravitational repulsion caused by the negative pressure will drive the region into a period of exponential expansion. If the energy density of the false vacuum is at the grand unified theory scale (, Eq. (10) shows that the time constant of the exponential expansion would be about sec, and that the corresponding Hubble length would be about cm. For inflation to achieve its goals, this patch has to expand exponentially for at least 65 efoldings, but the amount of inflation could be much larger than this. The exponential expansion dilutes away any particles that are present at the start of inflation, and also smooths out the metric. The expanding region approaches a smooth de Sitter space, independent of the details of how it began (Jensen & SteinSchabes, 1987). Eventually, however, the inflaton field at any given location will roll off the hill, ending inflation. When it does, the energy density that has been locked in the inflaton field is released. Because of the coupling of the inflaton to other fields, that energy becomes thermalized to produce a hot soup of particles, which is exactly what had always been taken as the starting point of the standard big bang theory before inflation was introduced. From here on the scenario joins the standard big bang description. The role of inflation is to establish dynamically the initial conditions which otherwise would have to be postulated.
The inflationary mechanism produces an entire universe starting from essentially nothing, so one would naturally want to ask where the energy for this universe comes from. The answer is that it comes from the gravitational field. The universe did not begin with this colossal energy stored in the gravitational field, but rather the gravitational field can supply the energy because its energy can become negative without bound. As more and more positive energy materializes in the form of an evergrowing region filled with a highenergy scalar field, more and more negative energy materializes in the form of an expanding region filled with a gravitational field. The total energy remains constant at some very small value, and could in fact be exactly zero. There is nothing known that places any limit on the amount of inflation that can occur while the total energy remains exactly zero.^{1}^{1}1In Newtonian mechanics the energy density of a gravitational field is unambiguously negative; it can be derived by the same methods used for the Coulomb field, but the force law has the opposite sign. In general relativity there is no coordinateinvariant way of expressing the energy in a space that is not asymptotically flat, so many experts prefer to say that the total energy is undefined. Either way, there is agreement that inflation is consistent with the general relativistic description of energy conservation.
Note that while inflation was originally developed in the context of grand unified theories, the only real requirements on the particle physics are the existence of a false vacuum state, and the possibility of creating the net baryon number of the universe after inflation.
3 Evidence for Inflation
Inflation is not really a theory, but instead it is a paradigm, or a class of theories. As such, it does not make specific predictions in the same sense that the standard model of particle physics makes predictions. Each specific model of inflation makes definite predictions, but the class of models as a whole can be tested only by looking for generic features that are common to most of the models. Nonetheless, there are a number of features of the universe that seem to be characteristic consequences of inflation. In my opinion, the evidence that our universe is the result of some form of inflation is very solid. Since the term inflation encompasses a wide range of detailed theories, it is hard to imagine any reasonable alternative.^{2}^{2}2The cyclicekpyrotic model (Steinhardt & Turok 2002) is touted by its authors as a rival to inflation, but in fact it incorporates inflation and uses it to explain why the universe is so large, homogeneous, isotropic, and flat.
The basic arguments for inflation are as follows:

The universe is big
First of all, we know that the universe is incredibly large: the visible part of the universe contains about particles. Since we have all grown up in a large universe, it is easy to take this fact for granted: of course the universe is big, it’s the whole universe! In “standard” FriedmannRobertsonWalker cosmology, without inflation, one simply postulates that about or more particles were here from the start. Many of us hope, however, that even the creation of the universe can be described in scientific terms. Thus, we are led to at least think about a theory that might explain how the universe got to be so big. Whatever that theory is, it has to somehow explain the number of particles, or more. One simple way to get such a huge number, with only modest numbers as input, is for the calculation to involve an exponential. The exponential expansion of inflation reduces the problem of explaining particles to the problem of explaining 60 or 70 efoldings of inflation. In fact, it is easy to construct underlying particle theories that will give far more than 70 efoldings of inflation. Inflationary cosmology therefore suggests that, even though the observed universe is incredibly large, it is only an infinitesimal fraction of the entire universe.

The Hubble expansion
The Hubble expansion is also easy to take for granted, since we have all known about it from our earliest readings in cosmology. In standard FRW cosmology, the Hubble expansion is part of the list of postulates that define the initial conditions. But inflation actually offers the possibility of explaining how the Hubble expansion began. The repulsive gravity associated with the false vacuum is just what Hubble ordered. It is exactly the kind of force needed to propel the universe into a pattern of motion in which each pair of particles is moving apart with a velocity proportional to their separation.

Homogeneity and isotropy
The degree of uniformity in the universe is startling. The intensity of the cosmic background radiation is the same in all directions, after it is corrected for the motion of the Earth, to the incredible precision of one part in 100,000. To get some feeling for how high this precision is, we can imagine a marble that is spherical to one part in 100,000. The surface of the marble would have to be shaped to an accuracy of about 1,000 angstroms, a quarter of the wavelength of light. Although modern technology makes it possible to grind lenses to quarterwavelength accuracy, we would nonetheless be shocked if we unearthed a stone, produced by natural processes, that was round to an accuracy of 1,000 angstroms.
The cosmic background radiation was released about 400,000 years after the big bang, after the universe cooled enough so that the opaque plasma neutralized into a transparent gas. The cosmic background radiation photons have mostly been traveling on straight lines since then, so they provide an image of what the universe looked like at 400,000 years after the big bang. The observed uniformity of the radiation therefore implies that the observed universe had become uniform in temperature by that time. In standard FRW cosmology, a simple calculation shows that the uniformity could be established so quickly only if signals could propagate at about 100 times the speed of light, a proposition clearly contradicting the known laws of physics.
In inflationary cosmology, however, the uniformity is easily explained. It is created initially on microscopic scales, by normal thermalequilibrium processes, and then inflation takes over and stretches the regions of uniformity to become large enough to encompass the observed universe and more.

The flatness problem
I find the flatness problem particularly impressive, because of the extraordinary numbers that it involves. The problem concerns the value of the ratio
(13) where is the average total mass density of the universe and is the critical density, the density that would make the universe spatially flat. (In the definition of “total mass density,” I am including the vacuum energy associated with the cosmological constant , if it is nonzero.)
For the past several decades there has been general agreement that lies in the range
(14) but for most of this period it was very hard to pinpoint the value with more precision. Despite the breadth of this range, the value of at early times is highly constrained, since is an unstable equilibrium point of the standard model evolution. Thus, if was ever exactly equal to one, it would remain exactly one forever. However, if differed slightly from one in the early universe, that difference—whether positive or negative—would be amplified with time. In particular, it can be shown that grows as
(15) At sec, for example, when the processes of big bang nucleosynthesis were just beginning, Dicke and Peebles (1979) pointed out that must have equaled one to an accuracy of one part in . Classical cosmology provides no explanation for this fact—it is simply assumed as part of the initial conditions. In the context of modern particle theory, where we try to push things all the way back to the Planck time, sec, the problem becomes even more extreme. If one specifies the value of at the Planck time, it has to equal one to 58 decimal places in order to be anywhere in the range of Eq. (14) today.
While this extraordinary flatness of the early universe has no explanation in classical FRW cosmology, it is a natural prediction for inflationary cosmology. During the inflationary period, instead of being driven away from one as described by Eq. (15), is driven towards one, with exponential swiftness:
(16) where is the Hubble parameter during inflation. Thus, as long as there is a sufficient period of inflation, can start at almost any value, and it will be driven to unity by the exponential expansion. Since this mechanism is highly effective, almost all inflationary models predict that should be equal to one (to within about 1 part in ). Until the past few years this prediction was thought to be at odds with observation, but with the addition of dark energy the observationally favored value of is now essentially equal to one. According to the latest WMAP results (Bennett et al. 2003), , in beautiful agreement with inflationary predictions.

Absence of magnetic monopoles
All grand unified theories predict that there should be, in the spectrum of possible particles, extremely massive particles carrying a net magnetic charge. By combining grand unified theories with classical cosmology without inflation, Preskill (1979) found that magnetic monopoles would be produced so copiously that they would outweigh everything else in the universe by a factor of about . A mass density this large would cause the inferred age of the universe to drop to about 30,000 years! Inflation is certainly the simplest known mechanism to eliminate monopoles from the visible universe, even though they are still in the spectrum of possible particles. The monopoles are eliminated simply by arranging the parameters so that inflation takes place after (or during) monopole production, so the monopole density is diluted to a completely negligible level.

Anisotropy of the cosmic microwave background (CMB) radiation
The process of inflation smooths the universe essentially completely, but density fluctuations are generated as inflation ends by the quantum fluctuations of the inflaton field. Several papers emerging from the Nuffield Workshop in Cambridge, UK, 1982, showed that these fluctuations are generically adiabatic, Gaussian, and nearly scaleinvariant (Starobinsky 1982; Guth & Pi 1982; Hawking 1982; Bardeen et al. 1983).^{3}^{3}3The concept that quantum fluctuations might provide the seeds for cosmological density perturbations, which goes back at least to Sakharov (1965), was pursued in the early 1980s by Lukash (1980a, 1980b), Press (1980, 1981), and Mukhanov & Chibisov (1981, 1982). Mukhanov & Chibisov’s papers are of particular interest, since they considered such quantum fluctuations in the context of the Starobinsky (1979, 1980) model, now recognized as a version of inflation. There is some controversy and ongoing discussion concerning the historical role of the Mukhanov & Chibisov papers, so I include a few comments that the reader can pursue if interested. Mukhanov & Chibisov first discovered that quantum fluctuations prevent the Starobinsky model from solving the initial singularity problem. They then considered the possibility that the quantum fluctuations are relevant for density perturbations, and found a nearly scaleinvariant spectrum during the de Sitter phase. Without any derivation that I can presently discern, the 1981 paper gives a nearly scaleinvariant formula for the final density perturbations after the end of inflation, which is similar but not identical to the result that was later described in detail in Mukhanov, Feldman, & Brandenberger (1992). In a recent preprint, Mukhanov (2003) refers to Mukhanov & Chibisov (1981) as “the first paper where the spectrum of inflationary perturbations was calculated.” But controversies surrounding this statement remain unresolved. Why, for example, were the authors never explicit about the subtle question of how they calculated the evolution of the (conformally flat) density perturbations in the de Sitter phase into the (conformally Newtonian) perturbations after reheating? This gap seems particularly evident in the longer 1982 paper. And could the Starobinsky model properly be considered an inflationary model in 1981 or 1982, since at the time there was no recognition in the literature that the model could be used to explain the homogeneity, isotropy, or flatness of the universe? It was not until later that Whitt (1984) and Mijic, Morris, & Suen (1986) established the equivalence between the Starobinsky model and standard inflation. After the 1981 and 1982 Mukhanov & Chibisov papers, the topic of density perturbations in the Starobinsky model was revisited by a number of authors, starting with Starobinsky (1983).
When my colleagues and I were trying to calculate the spectrum of density perturbations from inflation in 1982, I never believed for a moment that it would be measured in my lifetime. Perhaps the few lowest moments would be measured, but certainly not enough to determine a spectrum. But I was wrong. The fluctuations in the CMB have now been measured to exquisite detail, and even better measurements are in the offing. So far everything looks consistent with the predictions of the simplest, generic inflationary models. Fig. 3 shows the temperature power spectrum and the temperaturepolarization crosscorrelation, based on the first year of data of the WMAP experiment (Bennett et al. 2003). The curve shows the bestfit “runningindex” CDM model. The gray band indicates one standard deviation of uncertainty due to cosmic variance (the limitation imposed by being able to sample only one sky). The underlying primordial spectrum is modeled as a power law , where corresponds to scaleinvariance. The best fit to WMAP alone gives . When WMAP data is combined with data on smaller scales from other observations there is some evidence that grows with scale, but this is not conclusive. As mentioned above, the fit gives . The addition of isocurvature modes does not improve the fit, so the expectation of adiabatic perturbations is confirmed, and various tests for nonGaussianity have found no signs of it.
4 The Inflationary Power Spectrum
A complete derivation of the density perturbation spectrum arising from inflation is a very technical subject, so the interested reader should refer to the Mukhanov et al. (1992) or Liddle & Lyth (1993) review articles. However, in this section I will describe the basics of the subject, for single field slow roll inflation, in a simple and qualitative way.
For a flat universe () the metric of Eq. (3) reduces to
(17) 
The perturbations are described in terms of linear perturbation theory, so it is natural to describe the perturbations in terms of a Fourier expansion in the comoving coordinates . Each mode will evolve independently of all the other modes. During the inflationary era the physical wavelength of any given mode will grow with the scale factor , and hence will grow exponentially. The Hubble length , however, is approximately constant during inflation. The modes of interest will start at wavelengths far less than , and will grow during inflation to be perhaps 20 orders of magnitude larger than . For each mode, we will let (“first Hubble crossing”) denote the time at which the wavelength is equal to the Hubble length during the inflationary era. When inflation is over the wavelength will continue to grow as the scale factor, but the scale factor will slow down to behave as during the radiationdominated era, and during the matterdominated era. The Hubble length will grow linearly with , so eventually the Hubble length will overtake the wavelength, and the wave will come back inside the Hubble length. We will let (“second Hubble crossing”) denote the time for each mode when the wavelength is again equal to the Hubble length. This pattern of evolution is important to our understanding, because complicated physics can happen only when the wavelength is smaller than or comparable to the Hubble length. When the wavelength is large compared to the Hubble length, the distance that light can travel in a Hubble time becomes small compared to the wavelength, and hence all motion is very slow and the pattern is essentially frozen in.
Inflation ends when a scalar field rolls down a hill in a potential energy diagram, such as Figs. 1 or 2. Since the scalar field undergoes quantum fluctuations, however, the field will not roll homogeneously, but instead will get a little ahead in some places and a little behind in others. Hence inflation will not end everywhere simultaneously, but instead the ending time will be a function of position:
(18) 
Since some regions will undergo more inflation than others, we have a natural source of inhomogeneities.
Next we need to define a statistical quantity that characterizes the perturbations. Letting describe the fractional perturbation in the total energy density , useful Fourier space quantities can be defined by
(\thechapter.19a)  
(\thechapter.19b) 
where the brackets denote an expectation value.
Since the wave pattern is frozen when the wavelength is large compared to the Hubble length, for any given mode the pattern is frozen between and . We therefore expect a simple relationship between the amplitude of the perturbation at times and , where the perturbation at time is described by a time offset in the evolution of the scalar field, and at it is described by . Since we are approximating the problem with first order perturbation theory, the relationship must be linear. By dimensional analysis, the relationship must have the form
(20) 
where is a dimensionless constant and is the only quantity with units of inverse time that seems to have relevance. Of course deriving Eq. (20) and determining the value of is a lot of work.
To estimate , note that we expect its value to become frozen at about time . If the classical, homogeneous rolling of the scalar field down the hill is described by , then the offset in time is equivalent to an offset of the value of the scalar field,
(21) 
The sign is not very important, but it is negative because inflation will end earliest () in regions where the scalar field has advanced the most (, assuming ). is in principle calculable by solving Eq. (12), omitting the spatial Laplacian term. Although it is a 2nd order equation, for “slow roll” inflation one assumes that the term is negligible, so
(22) 
where GeV is the Planck mass. can be estimated by defining the quantity in analogy to Eqs. (19), but the quantity on the righthand side is just the scalar field propagator of quantum field theory. One can approximate as a free massless quantum field evolving in de Sitter space (see for example Birrell & Davies, 1982). We want to evaluate for . Again we can rely on dimensional analysis, since has the units of mass, and the only relevant quantity with dimensions of mass is . Thus , and Eqs. (20)–(22) can be combined to give
(23) 
where and are dimensionless constants, and . The entire quantity on the righthand side is evaluated at , since it is at this time that the amplitude of the mode is frozen.
Eq. (23) is the key result. It describes density perturbations which are nearly scale invariant, meaning that is approximately independent of , because typically and are nearly constant during the period when perturbations of observable wavelengths are passing through the Hubble length during inflation. Since is measurable and is calculable, one can use Eq. (23) to determine the value of . Using COBE data, Liddle and Lyth (1993) found
(24) 
While Eq. (23) describes density perturbations that are nearly scale invariant, it also allows us to express the departure from scale invariance in terms of derivatives of the potential . One defines the scalar index by
(25) 
so
(26) 
To carry out the differentiation, note that is related to by . Treating as a constant since it varies much more slowly than , differentiation gives . Using Eq. (22) for , one has (Liddle & Lyth 1992)
(27)  
where
(28) 
and are the now wellknown slowroll parameters that quantify departures from scale invariance. (But the reader should beware that some authors use slightly different definitions.) Alternatively, Eqs. (22) can be used to express and in terms of time derivatives of :
(29) 
so
(30) 
The above equation can be used to motivate a generic estimate of how much is likely to deviate from 1. Since inflation needs to end at roughly 60 efolds from the time when the righthand side of Eq. (23) is evaluated, we can take as the typical time scale for the variation of physical quantities. For any quantity we can generically estimate that , so will deviate from 1 by an amount of order 0.1. Of course any detailed model will make a precise prediction for the value of . . We can conclude that typically
5 Eternal Inflation: Mechanisms
The remainder of this article will discuss eternal inflation—the questions that it can answer, and the questions that it raises. In this section I discuss the mechanisms that make eternal inflation possible, leaving the other issues for the following sections. I will discuss eternal inflation first in the context of new inflation, and then in the context of chaotic inflation, where it is more subtle.
In the case of new inflation, the exponential expansion occurs as the scalar field rolls from the false vacuum state at the peak of the potential energy diagram (see Fig. 1) towards the trough. The eternal aspect occurs while the scalar field is hovering around the peak. The first model of this type was constructed by Steinhardt (1983), and later that year Vilenkin (1983) showed that new inflationary models are generically eternal. The key point is that, even though classically the field would roll off the hill, quantummechanically 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 (Guth & Pi 1985). The time scale for the decay of the false vacuum is controlled by
(31) 
the negative masssquared 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 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 firstorder phase transitions are round (Coleman & De Luccia 1980), the local universes formed in eternal new inflation are generally very irregular, as can be seen for example in the twodimensional simulation in Fig. 2 of Vanchurin, Vilenkin, & Winitzki (2000).)
In the context of chaotic inflationary models the situation is slightly more subtle. Andrei Linde (1986a, 1986b, 1990) showed that these models are eternal in 1986. In this case inflation occurs as the scalar field rolls down a hill of the potential energy diagram, as in Fig. 2, 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 (one Hubble time), in a region of one Hubble volume . Suppose that is the average value of 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 . (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 . One of the deep truths that one learns by working on inflation is that 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 Hubblesized region grows and breaks up into 20 independent Hubblesized regions.
As the scalar field is classically rolling down the hill, the classical change in the field during the time interval is going to be modified by quantum fluctuations , 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 during the interval by
(32) 
In lowest order perturbation theory the fluctuation is treated as a free quantum field, which implies that , 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 (Vilenkin & Ford 1982; Linde 1982b; Starobinsky 1982, 1986). There is then always some probability that the sum of the two terms on the righthand 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 will be larger at the end of the time interval 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 . For a Gaussian probability distribution, this condition will be met provided that the standard deviation for is bigger than . Using , the criterion becomes
(33) 
Comparing with Eq. (23), we see that the condition for eternal inflation is equivalent to the condition that on ultralong length scales is bigger than a number of order unity.
The probability that is positive tends to increase as one considers larger and larger values of , so sooner or later one reaches the point at which inflation becomes eternal. If one takes, for example, a scalar field with a potential
(34) 
then the de Sitter space equation of motion in flat RobertsonWalker coordinates (Eq. (17)) takes the form
(35) 
where spatial derivatives have been neglected. In the “slowroll” approximation one also neglects the term, so , where the Hubble constant is related to the energy density by
(36) 
Putting these relations together, one finds that the criterion for eternal inflation, Eq. (33), becomes
(37) 
Since must be taken very small, on the order of , 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
(38) 
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 it could possibly be avoided, 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 less rigorous, but I still feel confident that it is essentially correct. For eternal inflation to set in, all one needs is that the probability for the field to increase in a given Hubblesized volume during a Hubble time interval is larger than 1/20.
Thus, once inflation happens, it produces not just one universe, but an infinite number of universes.
6 Eternal Inflation: Implications
In spite of the fact that the other universes created by eternal inflation are too remote to imagine observing directly, I nonetheless claim that eternal inflation has real consequences in terms of the way we extract predictions from theoretical models. Specifically, there are three consequences of eternal inflation that I will discuss.
First, eternal inflation implies that all hypotheses about the ultimate initial conditions for the universe—such as the Hartle & Hawking (1983) no boundary proposal, the tunneling proposals by Vilenkin (1984, 1986, 1999) or Linde (1984, 1998), or the more recent Hawking & Turok (1998) instanton — become totally divorced from observation. That is, one would expect that if inflation is to continue arbitrarily far into the future with the production of an infinite number of pocket universes, then the statistical properties of the inflating region should approach a steady state which is independent of the initial conditions. Unfortunately, attempts to quantitatively study this steady state are severely limited by several factors. First, there are ambiguities in defining probabilities, which will be discussed later. In addition, the steady state properties seem to depend strongly on superPlanckian physics which we do not understand. That is, the same quantum fluctuations that make eternal chaotic inflation possible tend to drive the scalar field further and further up the potential energy curve, so attempts to quantify the steady state probability distribution (Linde, Linde, & Mezhlumian, 1994; GarciaBellido & Linde, 1995) require the imposition of some kind of a boundary condition at large . Although these problems remain unsolved, I still believe that it is reasonable to assume that in the course of its unending evolution, an eternally inflating universe would lose all memory of the state in which it started.
Even if the universe forgets the details of its genesis, however, I would not assume that the question of how the universe began would lose its interest. While eternally inflating universes continue forever once they start, they are apparently not eternal into the past. (The word eternal is therefore not technically correct—it would be more precise to call this scenario semieternal or futureeternal.) The possibility of a quantum origin of the universe is very attractive, and will no doubt be a subject of interest for some time. Eternal inflation, however, seems to imply that the entire study will have to be conducted with literally no input from observation.
A second consequence of eternal inflation is that the probability of the onset of inflation becomes totally irrelevant, provided that the probability is not identically zero. Various authors in the past have argued that one type of inflation is more plausible than another, because the initial conditions that it requires appear more likely to have occurred. In the context of eternal inflation, however, such arguments have no significance. Any nonzero probability of onset will produce an infinite spacetime volume. If one wants to compare two type of inflation, the expectation is that the one with the faster exponential time constant will always win.
A corollary to this argument is that new inflation is not dead. While the initial conditions necessary for new inflation cannot be justified on the basis of thermal equilibrium, as proposed in the original papers (Linde 1982; Albrecht & Steinhardt 1982), in the context of eternal inflation it is sufficient to conclude that the probability for the required initial conditions is nonzero. Since the resulting scenario does not depend on the words that are used to justify the initial state, the standard treatment of new inflation remains valid.
A third consequence of eternal inflation is the possibility that it offers to rescue the predictive power of theoretical physics. Here I have in mind the status of string theory, or the theory known as M theory, into which string theory has evolved. The theory itself has an elegant uniqueness, but nonetheless it appears that the vacuum is far from unique (Bousso & Polchinski 2000; Susskind 2003). Since predictions will ultimately depend on the properties of the vacuum, the predictive power of string/M theory may be limited. Eternal inflation, however, provides a possible mechanism to remedy this problem. Even if many types of vacua are equally stable, it may turn out that there is one unique metastable state that leads to a maximal rate of inflation. If so, then this metastable state will dominate the eternally inflating region, even if its expansion rate is only infinitesimally larger than the other possibilities. One would still need to follow the decay of this metastable state as inflation ends. It may very well branch into a number of final lowenergy vacua, but the number that are significantly populated could hopefully be much smaller than the total number of vacua. All of this is pure speculation at this point, because no one knows how to calculate these things. Nonetheless, it possible that eternal inflation might help to constrain the vacuum state of the real universe, perhaps significantly enhancing the predictive power of M theory.
7 Does Inflation Need a Beginning?
If the universe can be eternal into the future, is it possible that it is also eternal into the past? Here I will describe a recent theorem (Borde, Guth, & Vilenkin 2003) which shows, under plausible assumptions, that the answer to this question is no.^{4}^{4}4There were also earlier theorems about this issue by Borde & Vilenkin (1994, 1996) and Borde (1994), but these theorems relied on the weak energy condition, which for a perfect fluid is equivalent to the condition . This condition holds classically for forms of matter that are known or commonly discussed as theoretical proposals. It can, however, be violated by quantum fluctuations (Borde & Vilenkin 1997), and so the reliability of these theorems is questionable.
The theorem is based on the wellknown fact that the momentum of an object traveling on a geodesic through an expanding universe is redshifted, just as the momentum of a photon is redshifted. Suppose, therefore, we consider a timelike or null geodesic extended backwards, into the past. In an expanding universe such a geodesic will be blueshifted. The theorem shows that under some circumstances the blueshift reaches infinite rapidity (i.e., the speed of light) in a finite amount of proper time (or affine parameter) along the trajectory, showing that such a trajectory is (geodesically) incomplete.
To describe the theorem in detail, we need to quantify what we mean by an expanding universe. We imagine an observer whom we follow backwards in time along a timelike or null geodesic. The goal is to define a local Hubble parameter along this geodesic, which must be welldefined even if the spacetime is neither homogeneous nor isotropic. Call the velocity of the geodesic observer , where is the proper time in the case of a timelike observer, or an affine parameter in the case of a null observer. (Although we are imagining that we are following the trajectory backwards in time, is defined to increase in the future timelike direction, as usual.) To define , we must imagine that the vicinity of the observer is filled with “comoving test particles,” so that there is a test particle velocity assigned to each point along the geodesic trajectory, as shown in Fig. 4. These particles need not be real — all that will be necessary is that the worldlines can be defined, and that each worldline should have zero proper acceleration at the instant it intercepts the geodesic observer.
To define the Hubble parameter that the observer measures at time , the observer focuses on two particles, one that he passes at time , and one at , where in the end he takes the limit . The Hubble parameter is defined by
(39) 
where is the radial component of the relative velocity between the two particles, and 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.
The relative velocity between the observer and the test particles can be measured by the invariant dot product,
(40) 
which for the case of a timelike observer is equal to the usual special relativity Lorentz factor
(41) 
If is positive we would expect to decrease with , since we expect the observer’s momentum relative to the test particles to redshift. It turns out, however, that the relationship between and changes in can be made precise. If one defines
(42) 
then
(43) 
I like to call the “slowness” of the geodesic observer, because it increases as the observer slows down, relative to the test particles. The slowness decreases as we follow the geodesic backwards in time, but it is positive definite, and therefore cannot decrease below zero. corresponds to , or a relative velocity equal to that of light. This bound allows us to place a rigorous limit on the integral of Eq. (43). For timelike geodesics,
(44) 
where is the value of at the final time . For null observers, if we normalize the affine parameter by at the final time , then
(45) 
Thus, if we assume an averaged expansion condition, i.e., that the average value of the Hubble parameter along the geodesic is positive, then the proper length (or affine length for null trajectories) of the backwardsgoing geodesic is bounded. Thus the region for which is pastincomplete.
It is difficult to apply this theorem to general inflationary models, since there is no accepted definition of what exactly defines this class. However, in standard eternally inflating models, the future of any point in the inflating region can be described by a stochastic model (Goncharov, Linde, & Mukhanov 1987) for inflaton evolution, valid until the end of inflation. Except for extremely rare large quantum fluctuations, , where is the energy density of the false vacuum driving the inflation. The past for an arbitrary model is less certain, but we consider eternal models for which the past is like the future. In that case would be positive almost everywhere in the past inflating region. If, however, when averaged over a pastdirected geodesic, our theorem implies that the geodesic is incomplete.
There is of course no conclusion that an eternally inflating model must have a unique beginning, and no conclusion that there is an upper bound on the length of all backwardsgoing geodesics from a given point. There may be models with regions of contraction embedded within the expanding region that could evade our theorem. Aguirre & Gratton (2002, 2003) have proposed a model that evades our theorem, in which the arrow of time reverses at the hypersurface, so the universe “expands” in both halves of the full de Sitter space.
The theorem does show, however, that an eternally inflating model of the type usually assumed, which would lead to for pastdirected geodesics, cannot be complete. Some new physics (i.e., not inflation) would be needed to describe the past boundary of the inflating region. One possibility would be some kind of quantum creation event.
One particular application of the theory is the cyclic ekpyrotic model of Steinhardt & Turok (2002). This model has for null geodesics for a single cycle, and since every cycle is identical, when averaged over all cycles. The cyclic model is therefore pastincomplete, and requires a boundary condition in the past.
8 Calculation of Probabilities in Eternally Inflating Universes
In an eternally inflating universe, anything that can happen will happen; in fact, it will happen an infinite number of times. Thus, the question of what is possible becomes trivial—anything is possible, unless it violates some absolute conservation law. To extract predictions from the theory, we must therefore learn to distinguish the probable from the improbable.
However, as soon as one attempts to define probabilities in an eternally inflating spacetime, one discovers ambiguities. The problem is that the sample space is infinite, in that an eternally inflating universe produces an infinite number of pocket universes. The fraction of universes with any particular property is therefore equal to infinity divided by infinity—a meaningless ratio. To obtain a welldefined answer, one needs to invoke some method of regularization. In eternally inflating universes, however, the answers that one gets depend on how one chooses the method of regularization.
To understand the nature of the problem, it is useful to think about the integers as a model system with an infinite number of entities. We can ask, for example, what fraction of the integers are odd. With the usual ordering of the integers, , , , …, it seems obvious that the answer is . However, the same set of integers can be ordered by writing two odd integers followed by one even integer, as in . Taken in this order, it looks like of the integers are odd.
One simple method of regularization is a cutoff at equaltime surfaces in a synchronous gauge coordinate system. Specifically, suppose that one constructs a RobertsonWalker coordinate system while the model universe is still in the false vacuum (de Sitter) phase, before any pocket universes have formed. One can then propagate this coordinate system forward with a synchronous gauge condition,^{5}^{5}5By a synchronous gauge condition, I mean that each equaltime hypersurface is obtained by propagating every point on the previous hypersurface by a fixed infinitesimal time interval in the direction normal to the hypersurface. and one can define probabilities by truncating the spacetime volume at a fixed value of the synchronous time coordinate . I will refer to probabilities defined in this way as synchronous gauge probabilities.
An important peculiarity of synchronous gauge probabilities is that they lead to what I call the “youngness paradox”. The problem is that the volume of false vacuum is growing exponentially with time with an extraordinarily small time constant, in the vicinity of sec. Since the rate at which pocket universes form is proportional to the volume of false vacuum, this rate is increasing exponentially with the same time constant. This means that for every universe in the sample of age , there are approximately universes with age (1 sec). The population of pocket universes is therefore an incredibly youthdominated society, in which the mature universes are vastly outnumbered by universes that have just barely begun to evolve.
Probability calculations in this youthdominated ensemble lead to peculiar results, as was first discussed by Linde, Linde, & Mezhlumian (1995). Since mature universes are incredibly rare, it becomes likely that our universe is actually much younger than we think, with our part of the universe having reached its apparent maturity through an unlikely set of quantum jumps. These authors considered the expected behavior of the mass density in our vicinity, concluding that we should find ourselves very near the center of a spherical lowdensity region.
Since the probability measure depends on the method used to truncate the infinite spacetime of eternal inflation, we are not forced to accept the consequences of the synchronous gauge probabilities. A method of calculating probabilities that gives acceptable answers has been formulated by Vilenkin (1998) and his collaborators (Vanchurin, Vilenkin, & Winitzki, 2000; Garriga & Vilenkin, 2001). However, we still do not have a compelling argument from first principles that determines how probabilities should be calculated.
9 Conclusion
In this paper I have summarized the workings of inflation, and the arguments that strongly suggest that our universe is the product of inflation. I argued that inflation can explain the size, the Hubble expansion, the homogeneity, the isotropy, and the flatness of our universe, as well as the absence of magnetic monopoles, and even the characteristics of the nonuniformities. The detailed observations of the cosmic background radiation anisotropies continue to fall in line with inflationary expectations, and the evidence for an accelerating universe fits beautifully with the inflationary preference for a flat universe. Our current picture of the universe seems strange, with 95% of the energy in forms of matter that we do not understand, but nonetheless the picture fits together very well.
Next I turned to the question of eternal inflation, claiming that essentially all inflationary models are eternal. In my opinion this makes inflation very robust: if it starts anywhere, at any time in all of eternity, it produces an infinite number of pocket universes. Eternal inflation has the very attractive feature, from my point of view, that it offers the possibility of allowing unique (or possibly only constrained) predictions even if the underlying string theory does not have a unique vacuum. I discussed the past of eternally inflating models, concluding that under mild assumptions the inflating region must have a past boundary, and that new physics (other than inflation) is needed to describe what happens at this boundary. I have also described, however, that our picture of eternal inflation is not complete. In particular, we still do not understand how to define probabilities in an eternally inflating spacetime.
The bottom line, however, is that observations in the past few years have vastly improved our knowledge of the early universe, and that these new observations have been generally consistent with the simplest inflationary models.
Acknowledgments
This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DFFC0294ER40818.
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