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Proc. Natl. Acad. Sci. USA Vol. 90, pp. 4871-4877, June 1993 Colloquium Paper

This paper was presented at a coUoquium entitled "Physical ," organized by a committee chaired by David N. Schramm, held March 27 and 28, 1992, at the National Academy of Sciences, Irvine, CA. ALAN H. GUTH Center for Theoretical , Laboratory for Nuclear Science, and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139

ABSTRACT A short review of inflation is given, starting a box, letting the size of the box approach infinity at the end with a description of the underlying mechanism and the of the calculation. One can then think of the electromagnetic scenario of events. Four variations-old inflation, new infla- fields inside the box as a mechanical system. The can be tion, chaotic inflation, and extended inflation-are discussed. expanded as a sum of normal modes, and the coefficients of It is claimed that the inflationary model provides a plausible the normal mode functions can be taken as the dynamical explanation for (i) the large number ofparticles in the , degrees of freedom of the system. It turns out that each of (ii) the Hubble expansion, (iii) the large-scale uniformity ofthe these coefficients obeys the equations of a harmonic oscil- universe, (iv) the nearness of the universe to a critical density, lator, and there is no interaction between the coefficients. (v) the absence of magnetic monopoles, and (vi) the scale- The system is quantized by the same rules that one uses to invariant spectrum of microwave background fluctuations quantize the harmonic oscillator, and the result is that each observed by COBE (Cosmic Background Explorer). Finally, it normal mode has evenly spaced energy levels. In this case, is argued that the plausibility ofinflation is enhanced by the fact we interpret the nth excited state as the state that contains n that inflation is eternal. in the specified mode. Thus, the is inter- preted as the quantized excitation of a field. The inflationary model is a modification of the standard big In contemporary particle , all elementary particles are bang model that provides answers to a number of questions described in this way. There is an electron field to correspond that the fails to address. The inflationary to an electron, a quark field to correspond to a quark, a lasts for only a small fraction ofa second, and then the field to correspond to a neutrino, etc. Among the inflationary model joins smoothly onto the standard model. different types offields, the simplest is the -a field The role of inflation is to set up the initial conditions for the that has the same value to any Lorentz observer. The quan- standard model, conditions which otherwise would have to tized excitation ofa scalar field is a spinless particle. Although be assumed without explanation. Before inflation the total spinless particles that are regarded as elementary have yet to contained in the universe could have been as small as be observed, they are nonetheless akey ingredienttoa number 10 kg. If inflation is right, then the inflationary mechanism is of important . In particular, the Glashow-Weinberg- responsible for producing essentially all of the , en- Salam model of the electroweak interactions makes use of a ergy, and that exists in the universe today. scalar field, the Higgs field, to cause a symmetry in the theory In this lecture I will give a brief review of inflation, to be spontaneously broken. (If this symmetry were not describing the mechanism of inflation, the outline of the broken, then electrons and would both be massless scenario, the different types of inflationary models, and the and would be indistinguishable.) This Higgs field corresponds evidence for inflation. Finally I will briefly discuss the eternal to a neutral spinless Higgs particle, which will hopefully be of inflation: once inflation begins, calculations indi- observed at the SSC (Superconducting Supercoilider), if not cate that it ends locally but not globally. Although this before. Grand unified theories make use of similar Higgs property does not appear to lead to any observable predic- fields, but at a much higher mass scale, to spontaneously break tions, it certainly affects the way we think about the universe the grand unified symmetry that relates electrons, neutrinos, as a whole. and quarks. There is much current interest in superstring theories, I. The Mechanism of Inflation which actually go beyond the pattern ofthe field theories that were described above. In these theories the fundamental Before explaining the arguments for inflation, I will first object is not a field but is really a string-like object, which has explain how the inflationary model works. length but no width. These theories are believed to behave as The mechanism of inflation depends on scalar fields, so I field theories, however, at energy scales well below the will begin by briefly summarizing the role of scalar fields in scale (Mp 1/'G' = 1.02 x 1019 GeV, where G is . To begin with, the reader should recognize Newton's constant, and I use units for which A c 1; 1 that in the context of modern particle physics, all fundamen- GeV = 1.602 x 10-10 J). In fact, these field theories contain tal particles are described by fields. The best known example a large number of scalar fields. Thus, the particle physics is the photon. The classical equations describing the electro- motivation for believing that scalar fields exist is quite strong. magnetic field were written down in the 1860s, but then in the The electromagnetic field has a potential early 20th century physicists learned that the underlying laws given by V = (2 + A2)/81r, but the density of nature are and not classical. The quantization of of the scalar field has a wider range of possibilities. It is the electromagnetic field can be carried out in a very straight- restricted by the criterion that the field theory be renormal- forward way. For simplicity one can consider the fields inside izable (i.e., the requirement that the field theory leads ultimately to finite answers), but there are still several free The publication costs of this article were defrayed in part by page charge parameters involved in specifying the potential. The form of payment. This article must therefore be hereby marked "advertisement" the scalar field potential appropriate for the new inflationary in accordance with 18 U.S.C. §1734 solely to indicate this fact. universe model is shown in Fig. 1. The potential has a 4871 Downloaded by guest on October 1, 2021 4872 Colloquium Paper: Guth Proc. Natl. Acad. Sci. USA 90 (1993)

Before: False &,- FaLse Vacuum P=Pf , p=? (metastable)

{True Vacuum lP=o, p=O \True After: Vacuum _ -- ~dV

0 -OF 0 101 dW -pdV =pf dV FIG. 1. The potential energy function for the scalar field 4, appropriate for the new inflationary universe model. FIG. 2. A thought experiment to calculate the pressure of the false vacuum. As the piston chamber filled with false vacuum is minimum at a value of the scalar field that is nonzero, a enlarged, the energy density remains constant and the energy in- property that is generic for Higgs fields. The potential also creases. The extra energy is supplied by the agent puffing on the has a plateau centered at = 0, a feature that Higgs fields do piston, which must pull against the negative pressure of the false not necessarily have. This property is necessary, however, in vacuum. order for the new inflationary scenario to be possible. T.. -pf :.3, , [2] A particle physicist defines the word "vacuum" to mean the state of lowest possible energy density, so it is the state where gAl = diag[--1, 1, 1, 1] is the Lorentz . Thus, the in which lies at the minimum of the potential, labeled 4t on tensor of the false vacuum is Lorentz- the diagram. For emphasis, I will call this state the "true energy-momentum invariant, as one would expect, since the state is Lorentz- vacuum." Notice that the energy density of the true vacuum invariant. That is, if a region of has a scalar field with is shown as zero, which is equivalent to the statement that the a constant value = 0, to another Lorentz observer it will constant is either zero or small. cosmological immeasurably also look like a region of space with a scalar field of value Even if the constant has a value that is cosmo- cosmological = 0. One could in fact have used this Lorentz-invariance as logically significant, the effect on this diagram would be an alternative derivation of Eq. 1. energy completely imperceptible, since the scales of particle Finally, we are ready to discuss the gravitational effects of physics are so much higher. The explanation for the small- this very peculiar energy-momentum tensor. Starting with ness of the has been a long-standing the Einstein field equations mystery to particle physicists, although we now have at least a possible solution (1) in the context of a crude understanding 1 of . RILV g"vR = 87rGT1"' A~' [3] 2 The state in which the scalar field, in some region of space, is perched at the top of the plateau of the potential energy one sees that the effect of the energy-momentum tensor of diagram is called the "false vacuum." Note that the energy Eq. 2 is precisely the same as having a temporarily nonzero density of this state is pf and therefore is fixed by the laws of value of the cosmological constant A, related by physics. The false vacuum is of course not stable, since eventually the scalar field will no doubt evolve toward the A = 8trGpf. [4] minimum of the potential. Nonetheless, ifthe plateau is broad and flat enough, it will take a long for the scalar field to To see the effect on a Robertson-Walker universe, note that roll off the hill. Thus, the false vacuum is metastable and can the a(t) obeys the equation of motion, be very by the time scales of the early universe. long-lived 47r The crucial property of the false vacuum is that the energy a = --- G(p + 3p)a. [5] density is positive, but on short time scales it cannot be 3 lowered. This explains the etymology of the name "false vacuum." Here "false" is being used to mean temporary, In the present universe, the pressure term is a small relativ- and "vacuum" means the state of lowest possible energy istic correction. Ifthe universe were ever dominated by false density. vacuum, however, then the pressure term would have the It is now possible to construct a simple energy-conserva- opposite sign and overcome the gravitational attraction tion argument to determine the pressure of the false vacuum. caused by the energy density term: Imagine a chamber filled with false vacuum, as shown in Fig. 8ir 2. Since the energy density of the vacuum is fixed at pf, the a =- Gpfa. [6] energy inside the chamber is given by U = pfV, where V is 3 the volume ofthe chamber. Now suppose the piston is pulled outward, increasing the volume by dV. Unlike any normal The force of gravity actually becomes repulsive, and the substance, the false vacuum will maintain a constant energy expansion rate of the universe is accelerated. The general density despite the increase in volume. The change in energy solution to Eq. 6 is is then dU = pfdV, which must be equal to the work done, = + dW = -pdV, where p is the pressure. Thus, the pressure of a(t) clex' c2e-Xf, [7] the false vacuum is given by where cl and C2 are arbitrary constants, and the exponential rate is given by P =-4Pf. Lii

If one constructs the energy-momentum tensor corre- I8'T X= - Gpf. [8] sponding to this energy density and pressure, one finds 3 Downloaded by guest on October 1, 2021 Colloquium Paper: Guth Proc. Natl. Acad. Sci. USA 90 (1993) 4873 After some time the growing term will dominate, and the it enough and looks at only a small segment. The correlation expansion becomes a pure exponential. length for the scalar field is stretched by the expansion factor Thus, a scalar field in a false vacuum state can generate a to become at least about 10 cm. If the duration of inflation is repulsive gravitational force that sends the universe into a more than the minimal value, which seems quite likely, then period of exponential expansion. Such a scalar field is the final correlation length could be many orders of magni- generically called the . tude larger than 10 cm. There appears to be no upper limit to the amount of inflation that may have taken place. II. The Inflationary Universe Scenario The false vacuum is not stable, so it eventually decays by a process that can be regarded as a phase transition. When the The inflationary universe scenario begins with a patch of the phase transition takes place, the energy that has been stored universe somehow settling into a false vacuum state. The in the false vacuum is released in the form of new particles. mechanism by which this happens has no effect on the later These new particles rapidly come to thermal equilibrium, evolution. There are at least three different possibilities that resulting in a temperature with kT 1014 GeV. At this point have been discussed in the literature. the scenario rejoins the standard cosmological model. (i) Supercooling from high temperatures. This was the The are produced (see, e.g., ref. 22) by - earliest suggestion (2-4). If we assume that the universe nonconserving processes after inflation. Any baryons that began very hot, as is traditionally assumed in the standard big may have been present before inflation are simply diluted bang model, then as the universe cooled it presumably went away by the enormous expansion factor. Thus, inflationary through a number of phase transitions. For many types of cosmology requires an underlying particle theory, such as a scalar field potentials, supercooling into a false vacuum , in which the baryon number is not occurs naturally. This scenario has the difficulty, however, conserved. that there is no known mechanism to achieve the desired It is a dramatic feature of inflationary models that essen- preinflationary thermal equilibrium state. Calculations (5-8), tially all of the matter, energy, and entropy of the observed which will be discussed in Section III, show that the scalar universe is produced by the expansion and subsequent decay field must be very weakly coupled for quantum-induced of the false vacuum. (I used the qualifier "essentially" to density perturbations to be sufficiently small. Consequently, acknowledge the fact that a small patch of false vacuum is the scalar field would require much more than the available necessary time to relax to thermal equilibrium. This difficulty may not to start inflation. The minimal patch has a radius be however, because true thermal equilibrium is not of order X-1. For typical grand unified theory parameters, crucial, with a mass scale on the order a really necessary: a variety of random configurations give of 1014 GeV, the mass oa results that are very similar to those of thermal equilibrium minimal patch is of the order of 10 kg.) For this reason I (9-12). sometimes refer to the process of inflation as the ultimate free (ii) Tunneling from "nothing." These ideas are of course lunch. very speculative, since they involve a theory of quantum gravity that does not actually exist. However, the basic idea III. Varieties of Inflation seems very plausible. If is to be described by Inflation is not a precise theory but is rather a general idea quantum theory, then the geometry of space can presumably that can be implemented in a variety of ways. There are now undergo quantum transitions. One can then imagine an initial at least three distinct versions of inflationary models, all of state of absolute nothingness-the absence of matter, energy, which appear to be space, or time. The state of absolute nothingness can presum- workable. ably undergo a quantum transition to a small universe, which Old Inflation. The form of inflation proposed in the 1981 then forms the initial state for an inflationary scenario. Vari- paper (2) is now known as "old inflation," and it is not one ations ofthese ideas have been studied by Tryon (13), Vilenkin of the workable models. Old inflation contains a crucial flaw (14, 15), Linde (16-18), and Hartle and Hawking (19). that was discussed but not avoided in the original paper. The (iii) Random fluctuations in chaotic cosmology. In a ver- model was based on the hypothesis that the false vacuum sion called chaotic inflation, developed by Linde (20, 21), it corresponds to a local minimum of the potential energy is assumed that the scalar field 4 begins in a random state in diagram, as is illustrated in Fig. 3. A false vacuum of this sort which all 4) takes seems very plausible from the point of view of particle possible values of occur. Inflation then physics, and it is perfectly suitable for driving inflation. The place in those regions that have appropriate values of 4, and flaw becomes apparent when one calculates how inflation these inflated regions dominate the universe at later . would end, and it is therefore known as the "graceful exit" Regardless of which of the above mechanisms is assumed, problem. one expects that the correlation length of the scalar field just A false vacuum due to a local minimum of the before inflation is of the order of the inverse Hubble param- potential energy function would decay by the process of eter at the time, which is also of the order of the age of the Coleman-Callan (23, 24) bubble nucleation, a process that universe. For typical grand unified theory parameters, this resembles the way that water boils. Bubbles ofthe new phase gives a correlation length of about 10-24 cm. The patch then expands exponentially because of the gravitational repulsion of the false vacuum. To achieve the goals of inflation, we must assume that this exponential expansion results in an expansion factor ; 1025. For typical grand unified theory numbers, this enormous expansion requires only about 10-32 sec of inflation. During this infla- tionary period, the density of any particles that may have been present before inflation is diluted so much that it becomes completely negligible. At the same time, any non- uniformities in the metric of space are smoothed by the 0 enormous expansion. The explanation for this smoothness is identical to the reason why the surface of the appears to be flat, even though the earth is actually round-any FIG. 3. The potential energy function for the scalar field 4, as it differentiable curve looks like a straight line if one magnifies was assumed in the "old inflation" model. Downloaded by guest on October 1, 2021 4874 Colloquium Paper: Guth Proc. Natl. Acad. Sci. USA 90 (1993) would nucleate randomly in the background ofthe old phase, scale factor today, e is the wavelength today, and X is the and once formed these bubbles would grow until they col- exponential expansion rate during inflation, as in Eqs. 7 and lided with other bubbles. For the expected regime of param- 8. The perturbations are therefore nearly scale-invariant, eters, the bubble walls would move at a speed that rapidly varying only logarithmically with wavelength. This result is approaches the . In a , this kind certainly a significant for inflation, because a scale- of bubble growth would rapidly convert the universe to the invariant (Harrison-Zel'dovich) spectrum (27, 28) has for new phase, which is exactly what is needed to end inflation. some time appeared to be an attractive mechanism for The universe is not static, however, but continues to . expand in the regions that are not occupied by the bubbles. The magnitude of the perturbations, on the other hand, In fact, since the bubbles expand at essentially the speed of depends sensitively on the unknown parameter A and so light, implies that points outside the bubbles cannot cannot be predicted. For typical grand unified theory param- be influenced by the bubbles in any way. Thus, the expo- eters and typical galactic scales, Eq. 10 gives .p/p 102 VIk, nential expansion continues unhindered in the regions be- so that 8p/p S 10-5 implies A S 10-14. Thus, the tween the bubbles, making it very difficult for the bubbles to ofthe observed universe implies that the "inflaton field" must merge (25, 26). The energy of each bubble is concentrated in be extremely weakly coupled. This calculation implies that the the bubble wall, so the rarity of collisions implies that this inflaton field cannot be the Higgs field, as had been originally energy is not easily thermalized. In fact, it has been shown hoped. (26) that the bubbles would never merge (percolate) into an The severity ofthe constraint on A can be used to argue that infinite cluster but instead would continue indefinitely in the inflationary models look implausible, but such arguments clusters of finite size. Each typical cluster would be domi- have to be weighed against the successes ofinflation that will nated by a single largest bubble. The corresponding mass be discussed in the next section. In assessing the significance density would be grossly inhomogeneous, bearing no resem- of the A constraint, one should remember that there are a blance to the observed universe. number of very small dimensionless quantities in nature that The New Inflationary Universe. The first model to over- are not understood: the Yukawa coupling constant that come the graceful exit problem, producing a successful describes the interaction of the electron and the Higgs field example of inflation, was the new inflationary universe is about 10-6, and the ratio of the weak energy scale to the model, proposed independently by Linde (3) and Albrecht Planck scale is about 10-17. Clearly there are some funda- and Steinhardt (4). The exit problem is avoided by introduc- mental principles at work here that we are still missing. ing a scalar field potential with a flat plateau, as was shown Chaotic Inflation. While the new inflationary scenario in Fig. 1. This leads to a "slow-rollover" phase transition, in requires a potential energy function with a plateau, Linde (20, which quantum fluctuations destabilize the false vacuum, 21) has shown that such a severe restriction is not necessary. starting the scalar field to roll down the hill of the potential In a model known as chaotic inflation, Linde showed that energy diagram. These fluctuations are initially correlated inflation can work for a scalar field potential as simple as V(O) = A04, provided that one makes some mild assumptions over only a microscopic region, which is expected to be ofthe about the initial conditions. He proposed that the scalar field order of the inverse Hubble parameter. The accelerated begins in a chaotic state, so that there are some regions in expansion, however, continues as the scalar field begins to which the value of4 is a few times larger than the Planck mass slowly roll down the hill of the energy diagram. The addi- Mp. These regions must exceed some minimal size, which is tional inflation that takes place during the rolling can then estimated to be several times H-1, where H denotes the stretch the scale of correlation to be large enough to easily Hubble constant. Then 4 rolls down the hill of the potential encompass the observed universe. energy diagram, and a straightforward calculation indicates The rolling of the scalar field is subject to quantum fluc- that there is an adequate amount of inflation. The Hubble tuations, an effect that was first studied in detail by a group "constant" is not a constant in this case, but it is slowly of papers (5-8) that emerged from the Nuffield Workshop on varying, so the expansion can be called "quasi-exponential." the Very Early Universe, which took place in Cambridge, In the chaotic models it is not necessary for the scalar-field England, during the summer of 1982. These quantum fluctu- potential energy function V(O) to have a plateau, but as in ations imply that the rolling process is not completely uni- other models it must be very flat (i.e., weakly self-coupled) to form in space. The nonuniformities in the rolling develop into minimize the density perturbations that result from quantum perturbations in the mass density of the universe, which fluctuations. In particular, the constraint on the quartic cou- might be the primordial seeds from which the present struc- pling constant A is very similar to the constraint in new ture of the universe emerged. inflation. The perturbations are in principle calculable, but one needs Extended Inflation. A radically different solution to the to assume a form for the scalar-field energy function. One graceful exit problem has recently been developed by La and approach is to approximate the energy function near the Steinhardt (29) and is known as extended inflation. In this center of the plateau by the form model the effective gravitational constant G varies with time because of the nonminimal coupling of a scalar field to the 1 4 V(+)- V(0) - - A04, [9] scalar curvature R. The Lagrangian density for the theory can 4 be written as where A is a dimensionless coupling constant. The resulting 1 1 £T = -- R+2+ 2 - mass density perturbations are then found to be 8w 2 Oa VI() P0 IH3bbl /2g (aend 1 j~ - 0.2\k log1 -x II [10] + - ag,!OaJLcT- V2(u-) +... [11]-[ill Hubble ao where the left-hand side represents the fractional mass den- Here ao is the field that drives inflation in a manner that is sity fluctuations for a given wavelength, measured at the time essentially identical to old inflation. The potential energy when the physical wavelength is equal to the Hubble length. function for or would look much like Fig. 3. The field 4 is Here aend is the scale factor at the end of inflation, ao is the similar to the scalar field in the Brans-Dicke theory, and w is Downloaded by guest on October 1, 2021 Colloquium Paper: Guth Proc. Natl. Acad. Sci. USA 90 (1993) 4875

the Brans-Dicke parameter. [In a more elaborate version With inflation, however, we have a theory of the bang known as hyperextended inflation (30), the coupling between itself-the outward thrust of the can be attributed to and R is allowed to be more complicated.] The standard the repulsive gravity of the false vacuum. As we noticed in Lagrangian for Einstein gravity has the form T = R/(16irG), discussing Eq. 6, the false vacuum leads to a gravitational so a comparison shows that the effective value of Newton's repulsion that in turn leads to exponential expansion. This constant is given by G = co/(2Xr+2). During inflation the stable uniform expansion agrees precisely with the pattern of ex- solution to the evolution equations gives cx t (assuming that pansion discovered by Hubble. Vi is chosen to be negligible at these energies), which implies Homogeneity/. This argument is known as the that gravity is weakening with time. From Eq. 8 one can see homogeneity or isotropy problem, and it is also called the that this slows the expansion rate of the universe, and . The problem is that the extreme uniformity eventually the expansion slows to the point that the bubbles that is observed on very large scales, particularly in the of the new phase can fill the universe smoothly. Sufficiently cosmic background , cannot be explained in the standard cosmological model without inflation. rapid variation of the expansion rate requires (31, 32) that co s 25, which is inconsistent with astrophysical constraints on Observationally, the effective temperature of the cosmic background radiation is known to be isotropic to about one co in a pure Brans-Dicke theory. This problem can be evaded, however, by including a potential energy function If part in 103, and even this can be accounted for by Vi(O). back- V1(O) is large by the standards of available energy in the the assumption that the earth is moving through the universe today, then the present value of is essentially ground radiation. If one removes the dipole component that locked at the minimum of this function. The implications for can be attributed to the earth's motion, then the residual present experiments are then the same as if G were a anisotropy is at a level of about one part in 105 (33). This universal constant, as in Einstein gravity. extreme uniformity is very difficult to understand in the In extended inflation the inflaton field oc can have a context of the standard cosmological model, in which the potential energy function that is not strongly constrained, as horizon length (i.e., the distance that a light pulse could have long as it has a general shape similar to Fig. 3. In this model traveled since the initial singularity) is rather short. The it is the rolling of the *-field that controls how inflation ends, existence of horizons in cosmology was first discussed by so it must be extremely weakly coupled for the resulting Rindler (34), and the horizon problem was discussed by density fluctuations to be small. Weinberg (35) and by Misner et al. (36). An interesting feature of extended inflation is that the The cosmic background radiation that we now observe is resulting spectrum of density fluctuations is not scale- believed to have been emitted (or last scattered) at the time invariant but instead behaves as of recombination, about 105 years after the big bang, when the temperature was about 4000 K. (At earlier times the plasma that filled the universe was opaque to this 12 Pc°> e4l(2w-1) [12] radiation.) Thus, the measurements of the cosmic back- ground radiation give us effectively a snapshot of the uni- Hubble verse at an age of =z105 yr. Consider then two microwave For large this approaches the usual scale-invariant form, antennae pointing toward opposite directions in the sky. A but for smaller it is possible to obtain more power on larger simple calculation shows that the two detectors would be scales, which is approximately what one wants to reconcile detecting signals from sources that were separated from each current observations with theory. other, at the time of emission, by more than 90 horizon lengths (37). The problem is to understand how these two IV. Evidence for Inflation regions could possibly have come to be at the same temper- ature at the same time. There are at least six features of our universe that suggest that The horizon problem is not an inconsistency in the stan- it emerged from an inflationary beginning. dard model but represents instead a lack of explanatory Bigness of the Universe. To most students of cosmology, power. If the universe is assumed to have begun homoge- one of the most startling features of the universe is its neously, then it will continue to evolve homogeneously. The incredible size-the contains =1090 par- problem is that the very striking large-scale homogeneity of ticles. Since the standard big bang model (without inflation) the universe is not explained or predicted by the model but contains no mechanism to produce such a huge amount of instead must simply be assumed. entropy, the model requires us to assume that essentially all In the inflationary model, on the other hand, the horizon of these particles were here at the start. distance is stretched during inflation by a very large factor. The inflationary model, on the other hand, can actually At all subsequent times, the entire observed universe is much explain where such a vast number ofparticles can come from. smaller than the horizon distance. Furthermore, the infla- Particles are produced by the expansion and decay of the tionary model does more than simply enlarge the horizon false vacuum. Since the expansion is exponential, it makes distance-it actually provides a mechanism to create the sense to write 1090 = e207 = (e69)3. Thus, an exponential observed large-scale homogeneity. In the inflationary model expansion of69 e-foldings is sufficient to turn a single particle the size of the observable universe at times before the onset into 1090 particles. Therefore, inflation reduces the problem of inflation was smaller than it would have been in the of explaining the 1090 particles to the problem of explaining standard scenario by a factor 1025 or more. Thus, the entire why there were >69 e-foldings of inflation. In fact, it is easy observable universe would have been well within its horizon to construct underlying particle theories that will give far before inflation; it would have become homogeneous at this more than 69 e-foldings of inflation. The suggestion is that time by normal thermal processes, and then this very small even though the observed universe is incredibly large, it is homogeneous region would have been stretched by inflation only an infinitesimal fraction of the entire universe. to become large enough to encompass the observed universe. Hubble Expansion. Although the standard cosmological The region would then have remained homogeneous as it model is called the big bang, the theory in fact contains no continued to evolve. description whatever of the "bang." It is really a theory of Flatness. Inflation can also solve the "flatness" problem, the aftermath of a bang, describing how the matter expands which was first highlighted by Dicke and Peebles (38). That and cools, undergoes , and coagulates to is, inflation can explain why the mass density of the early form galaxies and other visible structures in the universe. universe was so close to the critical value. Downloaded by guest on October 1, 2021 4876 Colloquium Paper: Guth Proc. Natl. Acad. Sci. USA 90 (1993) The critical mass density Pc is defined as that mass density thing else by a factor of about 1012. A value this high would that is just barely sufficient to eventually halt the expansion imply that the expansion rate of the universe would slow to of the universe. Today the crucial ratio fl P/pc (where p is its present value in only about 30,000 years, which is rather the mass density of the universe) is known to lie somewhere clearly excluded by observation. in the range The monopole problem is easily solved in the context ofan inflationary model by arranging for the Higgs field to acquire 0.1 S Qf 2. [13] its nonzero expectation value either before or during the inflationary era. The monopole density would then become Despite the breadth ofthis range, the value offl at early times negligible as it is diluted by the inflation. Some monopoles is highly constrained, since fl = 1 is an unstable equilibrium would still have been produced by thermal fluctuations after point of the standard model evolution. Thus, if ft is exactly reheating, but this production is suppressed by a large equal to 1, it will remain exactly equal to 1 forever. If, on the Boltzmann factor-it would be negligible in the minimal other hand, fl were slightly greater than 1 in the early SU(5) model (46, 47) and presumably in most other models as universe, then it would have rapidly risen toward infinity. If well. it were slightly less than 1, it would have rapidly fallen toward Cosmic Background Explorer (COBE) Anisotropy Results. zero. In particular, it can be shown that ft - 1 grows as The nonuniformities in the cosmic microwave background have now been successfully measured by COBE, and the (during radiation-dominated era) results are in beautiful agreement with the scale-invariant ft {t2/3 (during matter-dominated era). spectrum that is expected from inflationary models. The measurements are quoted in terms of a spectral index n, with At t = 1 sec, for example, which was the beginning of the n = 1 corresponding to the scale-invariant spectrum. The processes of , must have been COBE measurements (33) yield n = 1.1 + 0.5. These mea- equal to 1 to an accuracy of 1 part in 1015. If we extrapolate surements are restricted by the 70 resolution of the instru- further to t = 1o-35 sec, the typical time scale for grand ment, which means that only very long wavelengths can be unified theory cosmology, we find that must have been probed. It is therefore useful to combine the COBE data with equal to 1 to an accuracy of 1 part in 1049. Standard cosmol- estimates for shorter wavelengths based on models of struc- ogy provides no explanation for this fact-it is simply as- ture formation. When the COBE team (48) carried out this sumed as part of the initial conditions. analysis, they found n = 1 + 0.23 for a span ofover 3 decades In the inflationary model, however, Q is driven during the in length scales. period of inflation very rapidly toward 1, as The COBE results are probably consistent with other mechanisms of inhomogeneity generation, such as cosmic Ql - 1 cx e-2xt, [15] strings, so the evidence is not conclusive. Nonetheless, it is important to remember that the inflationary model was where X is the Hubble constant during inflation. Thus, in an developed to explain the gross properties of the universe, inflationary model, one can begin with a value of ft far from such as the first 5 items on this list. It is therefore very unity, and inflation will drive ft toward 1 with spectacular impressive that the inflationary prediction for the fluctuation swiftness. This mechanism is so effective, in fact, that it is spectrum agrees perfectly with the observations. expected to overshoot by a wide margin. This leads to the cleanest prediction of inflation, which is that even today the V. The Eternal Nature of Inflation value of should differ from 1 by no more than about 1 part in 104 or 105. This discrepancy from 1 is a quantum effect, A fascinating feature of inflation, which in my opinion is also comprising the long wavelength tail of the density fluctua- important in evaluating the plausibility of inflation, is the fact tions that are possibly responsible for galaxy formation. that inflation is external-if inflation ever begins, then it will So far I have described the under the never stop (49-53). assumption that the cosmological constant A is identically To understand the endlessness of inflation, one first notices zero. If it is nonzero, then the role of inflation can be that the decay of the false vacuum, like the decay of many described by specifying that it drives the universe to a state other unstable systems, is an exponential process. S.-Y. Piand of geometric flatness, which corresponds to 1(54) have verified the exponential decay law in a simplified but exactly soluble model of a slow-rollover phase transition, A = - Qt + -H = 1. [16] in which the potential is taken as V(4) p2O2/2. For the case 3H2 of chaotic inflation, on the other hand, one might think that the scalar field would roll inexorably down the hill in the potential If A is regarded as a density, as in Eq. 4, then energy diagram, completing the decay in a finite time. Linde the A/(3H2) term can be interpreted as the vacuum energy (51) has shown, however, that if the scalar field starts at a contribution to Qttotal. With this definition, inflation always sufficiently high value, then it can be sustained by quantum drives the universe to fQtotal = 1. fluctuations, with again an exponential decay law. As long as Absence of Magnetic Monopoles. Another success of infla- the false vacuum endures, it drives an exponential expansion, tion is its ability to cure the " problem." and for reasonable parameters the rate of expansion is much In the context of grand unified theories, without faster than the rate of decay. Thus, even though the false inflation generally lead to huge excesses of superheavy vacuum is decaying, the total volume of the false vacuum (=1016 GeV) 't Hooft-Polyakov (39, 40; for a review, see ref. region actually increases with time. 41) magnetic monopoles. These monopoles are produced at As time goes on, pieces of the false vacuum region are the grand-unified-theory phase transition, when the grand- constantly undergoing decay. As each piece decays, it re- unified-theory Higgs fields acquire their nonzero values. The leases energy and thereby sets into motion a hot big bang rapidity of the phase transition implies that the correlation universe. However, other regions of false vacuum continue length of the Higgs fields is very short, and the fields to exponentially expand, so the false vacuum never disap- therefore (42) become tangled in a high density of knots- pears. these knots are the magnetic monopoles. For typical grand Since each hot big bang universe inflates to such an unified theories, the expected mass density of these magnetic enormous size, we would expect to be separated from the monopoles would exceed (43-45) the mass density of every- other by such immense distances that any contact Downloaded by guest on October 1, 2021 Colloquium Paper: Guth Proc. Natl. Acad. Sci. USA 90 (1993) 4877

would be impossible. It is even very likely that the space 8. Bardeen, J. M., Steinhardt, P. J. & Turner, M. S. (1983) Phys. distorts to such a degree that the other universes would Rev. D 28, 679-693. R. become disconnected. it can be argued 9. Albrecht, A., Brandenberger, R. & Matzner, (1985) Phys. causally Nonetheless, Rev. D 32, 1280-1289. that the eternal character of inflation makes it a more 10. Kung, J. & Brandenberger, R. H. (1990) Phys. Rev. D 42, plausible theory. In the absence of this feature, there is some 1008-1015. difficulty in deciding whether the initial conditions required 11. Brandenberger, R. H., Feldman, H. & Kung, J. (1991) Phys. for inflation are sufficiently plausible. Since there is no Scr. T36, 64-69. established theory of initial conditions, questions of this sort 12. Goldwirth, D. S. & Piran, T. (1992) Phys. Rep. 214, 223-291. can lead to inconclusive answers. Given the 13. Tryon, E. P. (1973) Nature (London) 246, 396-397. easily endless- 14. Vilenkin, A. 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