Inflation, Quantum Cosmology and the Anthropic Principle1 Andrei Linde Department of Physics, Stanford University, Stanford, CA 94305, USA 2 0 0 2 Contents v o N 1 Introduction 2 8 2 Chaotic inflation 3 2 v 3 Inflationary quantum fluctuations 7 8 4 4 Eternal chaotic inflation 8 0 1 1 5 Baby universes 12 2 0 6 From the universe to the multiverse 15 / h t - 7 Double universe model and the cosmological constant problem 18 p e h 8 Cosmological constant, dark energy, and the anthropic principle 21 : v i X 9 Problem of calculating the probabilities 23 r a 10 Does consciousness matter? 25 11 Why is mathematics so efficient? 27 12 Why quantum? 29 1 To appear in “Science and Ultimate Reality: From Quantum to Cosmos”, honoring John Wheeler’s90thbirthday. J.D.Barrow,P.C.W.Davies,&C.L.Harpereds. CambridgeUniversity Press (2003) 1 1 Introduction One of the main desires of physicists is to construct a theory that unambiguously predicts the observed values for all parameters of all elementary particles. It is very tempting to believe that the correct theory describing our world should be both beautiful and unique. However, most of the parameters of elementary particles look more like a col- lection of random numbers than a unique manifestation of some hidden harmony of Nature. For example, the mass of the electron is 3 orders of magnitude smaller than the mass of the proton, which is 2 orders of magnitude smaller than the mass of the W-boson, which is 17 orders of magnitude smaller than the Planck mass M . p Meanwhile, it was pointed out long ago that a minor change (by a factor of two or three) in the mass of the electron, the fine-structure constant α , the strong- e interaction constant α , or the gravitational constant G = M 2 would lead to a s p− universe in which life as we know it could never have arisen. Adding or subtract- ing even a single spatial dimension of the same type as the usual three dimensions would make planetary systems impossible. Indeed, in space-time with dimension- ality d > 4, gravitational forces between distant bodies fall off faster than r 2, and − in space-time with d < 4, the general theory of relativity tells us that such forces are absent altogether. This rules out the existence of stable planetary systems for d 6= 4. Furthermore, in order for life as we know it to exist, it is necessary that the universe be sufficiently large, flat, homogeneous, and isotropic. These facts, as well as a number of other observations, lie at the foundation of the so-called anthropic principle (Barrow and Tipler, 1986; Rozental, 1988; Rees, 2000) According to this principle, we observe the universe to be as it is because only in such a universe could observers like ourselves exist. Until very recently, many scientists were ashamed of using the anthropic prin- ciple in their research. A typical attitude was expressed in the book “The Early Universe” by Kolb andTurner: “It is unclear to oneof the authorshow a concept as lame as the “anthropic idea” was ever elevated to the status of a principle” (Kolb, 1990). This critical attitude is quite healthy. It is much better to find a simple physical resolutionoftheproblemratherthatspeculatethatwecanliveonlyintheuniverses wheretheproblemdoesnotexist. Thereisalwaysariskthattheanthropicprinciple does not cure the problem, but acts like a painkiller. On the other hand, this principle can help us to understand that some of the most complicated and fundamental problems may become nearly trivial if one looks at them from a different perspective. Instead of denying the anthropic principle or uncritically embracing it, one should take a more patient approach and check whether it is really helpful or not in each particular case. 2 There are two main versions of this principle: the weak anthropic principle and the strong one. The weak anthropic principle simply says that if the universe consists of different parts with different properties, we will live only in those parts where our life is possible. This could seem rather trivial, but one may wonder whether these different parts of the universe are really available. If it is not so, any discussion of altering the mass of the electron, the fine structure constant, and so forth is perfectly meaningless. The strong anthropic principle says that the universe must be created in such a way as to make our existence possible. At first glance, this principle must be faulty, because mankind, having appeared 1010 years after the basic features of our universe were laid down, could in no way influence either the structure of the universe or the properties of the elementary particles within it. Scientists oftenassociatedtheanthropicprinciplewiththeideathattheuniverse was created many times until the final success. It was not clear who did it and why was it necessary to make the universe suitable for our existence. Moreover, it would be much simpler to create proper conditions for our existence in a small vicinity of a solar system rather than in the whole universe. Why would one need to work so hard? Fortunately, most of the problems associated with the anthropic principle were resolved (Linde, 1983a,1984b,1986a)soonafter the invention of inflationary cosmol- ogy. Therefore we will remember here the basic principles of inflationary theory. 2 Chaotic inflation Inflationarytheory was formulatedinmany different ways, starting with themodels based on quantum gravity (Starobinsky, 1980) and on the theory of high temper- ature phase transitions with supercooling and exponential expansion in the false vacuum state (Guth, 1981; Linde, 1982a; Albrecht and Steinhardt, 1982). However, with the introduction of the chaotic inflation scenario (Linde, 1983b) it was real- ized that the basic principles of inflation actually are very simple, and no thermal equilibrium, supercooling, and expansion in the false vacuum is required. To explain the main idea of chaotic inflation, let us consider the simplest model ofascalarfieldφwithamassmandwiththepotentialenergydensityV(φ) = m2φ2, 2 see Fig. 1. Since this function has a minimum at φ = 0, one may expect that the scalar field φ should oscillate near this minimum. This is indeed the case if the universe does not expand. However, one can show that in a rapidly expanding universe the scalar field moves down very slowly, as a ball in a viscous liquid, viscosity being proportional to the speed of expansion. There are two equations which describe evolution of a homogeneous scalar field 3 in our model, the field equation φ¨+3Hφ˙ = −m2φ , (1) and the Einstein equation k 8π 1 H2 + = φ˙2 +V(φ) . (2) a2 3M2 2 p (cid:18) (cid:19) Here H = a˙/a is the Hubble parameter in the universe with a scale factor a(t) (the size of the universe), k = −1,0,1 for an open, flat or closed universe respectively, M is the Planck mass, M 2 = G, where G is the gravitational constant. The p p− first equation becomes similar to the equation of motion for a harmonic oscillator, ˙ where instead of x(t) we have φ(t). The term 3Hφ is similar to the term describing friction in the equation for a harmonic oscillator. If the scalar field φ initially was large, the Hubble parameter H was large too, according to the second equation. This means that the friction term was very large, and therefore the scalar field was moving very slowly, as a ball in a viscous liquid. Therefore at this stage the energy density of the scalar field, unlike the density of ordinarymatter,remainedalmostconstant,andexpansionoftheuniversecontinued with a much greater speed than in the old cosmological theory. Due to the rapid growth of the scale of the universe and a slow motion of the field φ, soon after the beginning of this regime one has φ¨≪ 3Hφ˙, H2 ≫ k , φ˙2 ≪ m2φ2, so the system of a2 equations can be simplified: a˙ 3 φ˙ = −m2φ , (3) a a˙ 2mφ π H = = . (4) a M 3 p r The last equation shows that the size of the universe a(t) in this regime grows approximately as eHt, where H = 2mφ π . Mp 3 q This stage of exponentially rapid expansion of the universe is called inflation. In realistic versions of inflationary theory its duration could be as short as 10 35 sec- − onds. Whenthefieldφbecomessufficiently small, viscosity becomes small, inflation ends, and the scalar field φ begins to oscillate near the minimum of V(φ). As any rapidly oscillating classical field, it looses its energy by creating pairs of elementary particles. These particles interact with each other and come to a state of thermal equilibrium with some temperature T. From this time on, the corresponding part of the universe can be described by the standard hot universe theory. The main difference between inflationary theory and the old cosmology becomes clear when one calculates the size of a typical inflationary domain at the end of inflation. Investigation of this issue shows that even if the initial size of inflationary 4 V Planck density A B C (cid:30) Figure 1: Motion of the scalar field in the theory with V(φ) = m2φ2. Several 2 different regimes are possible, depending on the value of the field φ. If the potential energy density of the field is greater than the Planck density ρ ∼ M4 ∼ 1094 g/cm3, p quantum fluctuations of space-time are so strong that one cannot describe it in usual terms. Such a state is called space-time foam. At a somewhat smaller energy density (region A: mM3 < V(φ) < M4) quantum fluctuations of space-time are p p small, but quantum fluctuations of the scalar field φ may be large. Jumps of the scalarfieldduetoquantumfluctuationsleadtoaprocessofeternalself-reproduction of inflationary universe which we are going to discuss later. At even smaller values of V(φ) (region B: m2M2 < V(φ) < mM3 ) fluctuations of the field φ are small; it p p slowly moves down as a ball in a viscous liquid. Inflation occurs both in the region A and region B. Finally, near the minimum of V(φ) (region C) the scalar field rapidly oscillates, creates pairs of elementary particles, and the universe becomes hot. 5 universe was as small as the Plank size l ∼ 10 33 cm, after 10 35 seconds of P − − inflation the universe acquires a huge size of l ∼ 101012 cm. This makes our universe almostexactlyflatandhomogeneousonlargescalebecauseallinhomogeneitieswere stretched by a factor of 101012. This number is model-dependent, but in all realistic models the size of the universe after inflation appears to be many orders of magnitude greater than the sizeofthepartoftheuniverse whichwecanseenow, l ∼ 1028 cm. Thisimmediately solves most of the problems of the old cosmological theory (Linde, 1990a). Consider a universe which initially consisted of many domains with chaotically distributed scalar field φ (or if one considers different universes with different values of the field). Those domains where the scalar field was too small never inflated, so they do not contribute much to the total volume of the universe. The main contribution to the total volume of the universe will be given by those domains which originally contained large scalar field φ. Inflation of such domains creates huge homogeneous islands out of the initial chaos, each homogeneous domain being much greater than the size of the observable part of the universe. That is why I called this scenario ‘chaotic inflation’. There is a big difference between this scenario and the old idea that the whole universe was created at the same moment of time (Big Bang), in a nearly uniform state with indefinitely large temperature. In the new theory, the condition of uni- formity and thermal equilibrium is no longer required. Each part of the universe could have a singular beginning (see (Borde et al, 2001) for a recent discussion of this issue). However, in the context of chaotic inflation, this does not mean that the universe as a whole had a single beginning. Different parts of the universe could come to existence at different moments of time, and then grow up to the size much greater than the total size of the universe. The existence of initial singularity (or singularities) does not imply that the whole universe was created simultaneously in a single Big Bang explosion. In other words, we cannot tell anymore that the whole universe was born at some time t = 0 before which it did not exist. This conclusion is valid for all versions of chaotic inflation, even if one does not take into account the process of self-reproduction of the universe discussed in Section 4. The possibility that our homogeneous part of the universe emerged from the chaotic state initial state has important implications for the anthropic principle. Until now we have considered the simplest inflationary model with only one scalar field. Realistic models of elementary particles involve many other scalar fields. For example, according to the standard theory of electroweak interactions, masses of all elementary particles depend on the value of the Higgs scalar field ϕ in our universe. This value is determined by the position of the minimum of the effective potential V(ϕ) for the field ϕ. In the simplest models, the potential V(ϕ) has only one minimum. However, in general, the potential V(ϕ) may have many different minima. For example, in the simplest supersymmetric theory unifying weak, strong 6 andelectromagneticinteractions, theeffectivepotentialhasseveraldifferentminima of equal depth with respect to the two scalar fields, Φ and ϕ. If the scalar fields Φ and ϕ fall to different minima in different parts of the universe (the process called spontaneous symmetry breaking), the masses of elementary particles and the laws describing their interactions will be different in these parts. Each of these parts may become exponentially large because of inflation. In some of these parts, there will be no difference between weak, strong and electromagnetic interactions, and life of our type will be impossible there. Some other parts will be similar to the one where we live now (Linde, 1983c). This means that even if we will be able to find the final theory of everything, we will be unable to uniquely determine properties of elementary particles in our universe; the universe may consist of different exponentially large domains where the properties of elementary particles may be different. This is an important step towards the justification of the anthropic principle. A further step can be made if one takes into account quantum fluctuations produced during inflation. 3 Inflationary quantum fluctuations According to quantum field theory, empty space is not entirely empty. It is filled with quantum fluctuations of all types of physical fields. The wavelengths of all quantumfluctuationsofthescalarfieldφgrowexponentiallyduringinflation. When the wavelength of any particular fluctuation becomes greater than H 1, this fluc- − tuation stops oscillating, and its amplitude freezes at some nonzero value δφ(x) ˙ because of the large friction term 3Hφ in the equation of motion of the field φ. The amplitude of this fluctuation then remains almost unchanged for a very long time, whereas its wavelength grows exponentially. Therefore, the appearance of such a frozen fluctuation is equivalent to the appearance of a classical field δφ(x) produced from quantum fluctuations. Because the vacuum contains fluctuations of all wavelengths, inflation leads to the continuous creation of new perturbations of the classical field with wavelengths greater than H 1. An average amplitude of perturbations generated during a time − intervalH 1 (inwhichtheuniverseexpandsbyafactorofe)isgivenby|δφ(x)| ≈ H − 2π (Vilenkin and Ford, 1982; Linde, 1982c). These quantum fluctuations are responsible for galaxy formation (Mukhanov and Chibisov, 1981; Hawking, 1982; Starobinsky, 1982; Guthand Pi, 1982; Bardeen et al, 1983). But if the Hubble constant during inflation is sufficiently large, quan- tum fluctuations of the scalar fields may lead not only to formation of galaxies, but also to the division of the universe into exponentially large domains with different properties. 7 As an example, consider again the simplest supersymmetric theory unifying weak, strong and electromagnetic interactions. Different minima of the effective potential in this model are separated from each other by the distance ∼ 10 3M . − p The amplitude of quantum fluctuations of the fields φ, Φ and ϕ in the beginning of chaotic inflation can be as large as 10 1M . This means that at the early stages of − p inflation the fields Φ and ϕ could easily jump from one minimum of the potential to another. Therefore even if initially these fields occupied the same minimum all over the universe, after the stage of chaotic inflation the universe becomes divided into many exponentially large domains corresponding to all possible minima of the effective potential (Linde, 1983c, 1984b). 4 Eternal chaotic inflation The process of the division of the universe into different parts becomes even easier if one takes into account the process of self-reproduction of inflationary domains. The basic mechanism can be understood as follows. If quantum fluctuations are sufficiently large, they may locally increase the value of the potential energy of the scalar field in some parts of the universe. The probability of quantum jumps leading to a local increase of the energy density can be very small, but the regions where it happens start expanding much faster than their parent domains, and quantum fluctuations inside them lead to production of new inflationary domains which expand even faster. This surprising behavior leads to the process of self- reproduction of the universe. This process is possible in the new inflation scenario (Steinhardt, 1982; Linde, 1982a; Vilenkin, 1983). However, even though the possibility to use this result for the justification of the anthropic principle was mentioned in (Linde, 1982a), this observation did not attract much attention because the amplitude of the fluctua- tions in new inflation typically is smaller than 10 6M . This is too small to probe − p most of the vacuum states available in the theory. As a result, the existence of the self-reproduction regime in the new inflation scenario was basically forgotten; for many years this effect was not studied or used in any way even by those who have found it. Thesituationchangeddramaticallywhenitwasfoundthattheself-reproduction of the universe occurs not only in new inflation but also in the chaotic inflation scenario (Linde, 1986a). In order to understand this effect, let us consider an inflationary domain of initial radius H 1 containing a sufficiently homogeneous − field with initial value φ ≫ M . Equations (3), (4) tell us that during a typical p time interval ∆t = H 1 the field inside this domain will be reduced by ∆φ = Mp2. − 4πφ Comparing this expression with the amplitude of quantum fluctuations δφ ∼ H = 2π 8 mφ , one can easily see that for φ ≫ φ ∼ Mp Mp, one has |δφ| ≫ |∆φ|, i.e. the √3πMp ∗ 2 m motion of the field φ due to its quantum fluctuaqtions is much more rapid than its classical motion. During the typical time H 1 the size of the domainof initial size H 1 containing − − the field φ ≫ φ grows e times, its volume increases e3 ∼ 20 times, and almost in a ∗ half of this new volume the field φ jumps up instead of falling down. Thus the total volume of inflationary domains containing the field φ ≫ φ grows approximately 10 ∗ times. During the next time interval H 1 this process continues; the universe enters − an eternal process of self-reproduction. I called this process ‘eternal inflation.’ In this scenario the scalar field may wander for an indefinitely long time at the density approaching the Planck density. This induces quantum fluctuations of all other scalar field, which may jump from one minimum of the potential energy to another for an unlimited time. The amplitude of these quantum fluctuations can be extremely large, δϕ ∼ δΦ ∼ 10 1M . As a result, quantum fluctuations − p generated during eternal chaotic inflation can penetrate through any barriers, even if they have Planckian height, and the universe after inflation becomes divided into indefinitely large number of exponentially large domains containing matter in all possible states corresponding to all possible mechanisms of spontaneous symmetry breaking, i.e. to the different laws of the law-energy physics (Linde, 1986a; Linde et al, 1994). Arich spectrum ofpossibilities may appear during inflationin Kaluza-Kleinand superstringtheories,whereanexponentiallylargevarietyofvacuumstatesandways of compactification is available for the original 10- or 11-dimensional space. The type of compactification determines coupling constants, vacuum energy, symmetry breaking, and finally, the effective dimensionality of the space we live in. As it was shown in (Linde and Zelnikov, 1988), chaotic inflation at a nearly Planckian density may lead to a local change of the number of compactified dimensions; the universe becomes divided into exponentially large parts with different dimensionality. Sometimes one may have a continuous spectrum of various possibilities. For ex- ample, in the context of the Brans-Dicke theory, the effective gravitational constant is a function of the Brans-Dicke field, which also experienced fluctuations during inflation. As a result, the universe after inflation becomes divided into exponen- tially large parts with all possible values of the gravitational constant G and the amplitude of density perturbations δρ (Linde, 1990b; Garcia-Bellido et al 1994). ρ Inflation may divide our universe into exponentially large domains with continu- ously varying baryon to photon ratio nB (Linde, 1985) and with galaxies having nγ vastly different properties (Linde, 1987b). Inflation may also continuously change the effective value of the vacuum energy (the cosmological constant Λ), which is a pre-requisite for many attempts to find an anthropic solution of the cosmological constant problem (Linde, 1984b,1986b; Weinberg, 1987; Efstathiou, 1995; Vilenkin, 9 1995b; Martel et al, 1998; Garriga and Vilenkin, 2000,2001b,2002; Bludman and Roos, 2002; Kallosh and Linde, 2002). Under these circumstances, the most diverse sets of parameters of particle physics (masses, coupling constants, vacuum energy, etc.) can appear after inflation. To illustrate the possible consequences of such theories in the context of infla- tionary cosmology, we present here the results of computer simulations of evolution of a system of two scalar fields during chaotic inflation (Linde et al, 1994). The field φ is the inflaton field driving inflation; it is shown by the height of the distribution of the field φ(x,y) in a two-dimensional slice of the universe. The field χ determines the type of spontaneous symmetry breaking which may occur in the theory. We paint the surface black if this field is in a state corresponding to one of the two minima of its effective potential; we paint it white if it is in the second minimum corresponding to a different type of symmetry breaking, and therefore to a different set of laws of low-energy physics. In the beginning of the process the whole inflationary domain was black, and the distribution of both fields was very homogeneous. Then the domain became exponentially large and it became divided into exponentially large domains with different properties, see Fig. 2. Each peak of the ‘mountains’ corresponds to a nearly Planckian density and can de interpreted as a beginning of a new Big Bang. The laws of physics are rapidly changing there, but they become fixed in the parts of the universe where the field φ becomes small. These parts correspond to valleys in Fig. 2. Thus quantum fluctuations of the scalar fields divide the universe into exponentially large domains with different laws of low-energy physics, and with different values of energy density. As a result of quantum jumps of the scalar fields during eternal inflation, the universe becomes divided into infinitely many exponentially large domains with different laws of low-energy physics. Each of these domains is so large that for all practical purposes it can be considered a separate universe: Its inhabitants will live exponentially far away from its boundaries, so they will never know anything about the existence of other ‘universes’ with different properties. If this scenario is correct, then physics alone cannot provide a complete expla- nation for all properties of our part of the universe. The same physical theory may yield large parts of the universe that have diverse properties. According to this scenario, we find ourselves inside a four-dimensional domain with our kind of physical laws not because domains with different dimensionality and with alternate properties are impossible or improbable, but simply because our kind of life cannot exist in other domains. This provides a simple justification of the weak anthropic principle and removes thestandardobjectionsagainst it. Onedoesnotneedanymore toassume thatsome supernatural cause created our universe with the properties specifically fine-tuned 10