Particle Physics and Inflationary Cosmology

Andrei D. Linde  Particle Physics and Inflationary Cosmology in Physics Today  (Sept. 1987)

   Elementary particle physics has developed towards unified theories of strong, weak, electromagnetic and gravitational interactions. In 1983 the Glashow-Weinberg-Salam theory which unified the weak and electromagnetic interactions made a very good start. In this way to reach our final goals, that is, verify experimentally the "theory of everything", we need a huge collider which should accelerate particles beyond about 1019 GeV. But the largest accelerator ring which can be built on Earth, with a circumference of 40,000 km, may not accelerate particles about 108 GeV. So we should try some different way from making a huge accelerator to find the unified theories.

1. The Universe as Accelerator

The only accelerator that could produce particles energy enough for a direct testing of the unified theories of all fundamental interactions is our universe itself! The Big Bang scenario - 'the hot-universe theory' assert that universe was born at some moment 𝑡 = 0 about 15 billion years ago, in a state of infinitely high temperature 𝑇 and infinitely large density 𝜌. (We will use a convenient unit system that sets 𝜅, 𝑐 and 𝘩 all equal to 1, so that 𝑙𝑝 equals 1/𝑀𝑝 and Planck density is 𝑀𝑝4, roughly 1094 g/cm3.) It is just at such time when all the unity of all four fundamental interactions would have been manifest.
   With the rapid expansion of the universe, the average energy of particle decrease rapidly, and the universe becomes cold. The temperature falls as the reciprocal of 𝑅, the scale factor, or "radius" of he universe. This means that particle interactions at extremely large energies can have occurred only at the very early stage of universal evolution. Thus it came as a great surprise to those who study elementary particles that the investigation of physical process at the very early stages of the universe can rule out most of the existing unified theories.
   For example, all the grand unified theories predicted the existence of superheavy stable particle carrying magnetic charge: monopoles. The standard hot-universe theory assert that monopoles should appear at the very early stage, and they should now be as abundant as protons. But the mean density of the matter would be about 15 orders of magnitude higher than its present value of about 10-29g/cm3. Also to be fair, one must say no consistent cosmological model based on  superstring theories has been suggested thus far.
   We see that it is not difficult to obtain strong cosmological constraints on the unified particle theories. On the contrary, there is a problem as to whether it is possible to reconcile elementary particle theory and cosmology. To answer this problem it is necessary to check whether the standard hot-universe theory is as good as as it is seemed at first to be, and whether it is possible to modify it so as to remove some of the difficulties we have found.

2. Problems of the Standard Theory

There are many difficult problems associated with the hot-universe theory. The most important problems are:
... The singularity problem.   The state of infinite density in which the universe was born at time 0 is called a singularity. What was before the Big Bang singularity? If space-time itself does not exist for time less than 0, how could everything appear from noting?
... The flatness problem.   According to general relativity, the geometry of the universe is different from the Euclidean and it may be closed in the way that the surface of a sphere is closed. But we see that our universe is just about flat on a scale of 1028cm, the radius of the observable part of the universe. This 1010-light-year distance to the Big Bang horizon is 60 order of magnitude larger than the Planck length 𝑙 𝑝. Why is our universe so flat, its geometry almost exactly Euclidean?
... The horizontal problem.   Considering an infinite (open or flat) universe: Why did all the causally disconnected regions of the infinite universe start their expansion simultaneously (at 𝑡 = 0)?
... The problem of homogeneity and galaxy formations   Astronomical observations shows that our universe on the very large scale is extremely homogeneous. On the scale of 1010 light years the distribution  of matter departs from perfect homogeneity by less than a part in thousand. why is the universe so homogeneous? On much smaller scale, the universe contains stars, galaxies, clusters of galaxies, voids and other structure larger than 108 light years. What is the origin of these important inhomogeneities?
... The uniqueness of homogeneity and galaxy formations  The essence of this problem was once formulated by Albert Einstein:"What I am really interested  in is whether God could have created the world differently." Though the Kaluza-Klein and superstring theories asserts that space-time originally has considerably more than four dimensions, and the extra dimensions have been "compactified," shrunk into thin tubes, why the compactification stopped with four effective space-time dimensions, not some other number? Moreover, in superstring theories there are thousands of possible compactification schemes to four dimensions, each giving different low energy particle physics and different vacuum energy density. Also in supersymmetric grand unified theories, for example, there are dozens of stable vacuum states, each corresponding to a different type of symmetry braking between fundamental interactions, and thus to a different phenomenology of low-energy particle physics. Why the compacttification and symmetry breaking brought the world we know, and no other.
   All these problems are very difficult within the standard hot-universe theory. But most of these problems, one after another, have been completely resolved or considerably relaxed in the context of one comparatively simple scenario of the evolution of the universe-the inflationary scenario.

3. The Inflationary Paradigm

There are many different versions of the inflationary -universe scenario. Sometime called the inflationary paradigm of all these versions, is the existence of some stage of evolution at which the universe expands (quasi-)exponentially while it is in a vacuumlike state containing some homogeneous classical fields, but almost no particles. Such an expansion is the inflation we refer to. After inflation on it can be described by usual hot-universe theory. (see Fig.1.1)
   Historically, the possible existence of an inflationary stage was first envisaged by Erast Gliner since 1965. In the 1970s Andrei Linde realized that homogeneous classical scalar fields 𝜑 can play the role of an unstable vacuum state, and that their decay can heat up the universe. In 1979 Alexey Starobinsky point out that the regime of exponential expansion and subsequent reheating of the universe occurs in the theory of gravity with quantum corrections.
   The crucial step in the inflationary paradigm was taken by Alan Guth in 1980. He suggested the stage of exponential expansion in some supercooled vacuumlike state to solve the primordial monopole problem, the flatness problem and the horizontal problem simultaneously. But the universe after inflation in his scenario becomes too inhomogenous.
   In October1981 Andrei Linde suggested an improved version of the inflation idea, whicj is called the new inflationary scenario. This resolved some of the difficulties of Guth's original formulation. Some moth later the same idea was proposed also by Andrea Albrecht and Paul Steinhart, and soon the new inflationary scenario became quite popular. However , it turns out to be very difficult to realize this scenario in the context of realistic theories of elementary particles. Moreover, the orthodox version was just a modest variation on the standard hot Big Bang theory. It was still assumed that there was an initial singularity at 𝑡 = 0, that after the Planck time 𝑡𝑝 (about 10-43 seconds) the universe became hot, and that inflation was just a brief interlude in the evolution of the universe. 
      In 1983 another inflationary scenario-the chaotic inflation scenario was proposed by A. Linde. In his opinion, this scenario is much simpler and more natural that other versions of inflation. Therefore he shall concentrate here on the chaotic inflation and some of its recent extents,which have given rise to the idea of an eternally existing chaotic inflationary universe. A more technical discussion of various versions of inflation can be found in a number of review articles and some books by A. Linde, S. Hawking, W. Israle, R. Brandenberger, M. Turner, A. Guth, and others in 1980s.  

4. Chaotic Inflation

The unified particle theories contain some scalar fields 𝜑 in addition to the spinor field 𝜓 describing electrons, neutrinos and quarks, and the vector fields 𝐴𝜇 mediating their interaction. Scalar fields have long been used for pions and its role in the new theories is more fundamental and complicated. If the potential energy 𝑉(𝜑) of a scalar field 𝜑 has one minimum at some value 𝜑1, then the whole universe gradually become filled with the field 𝜑1. Such a field is almost unobservable; it looks the same to a moving observer as it does to an observer at rest like the apprearence of empty space.
   However, the field gives different masses to vector particles mediating different interactions, such scalar fields, referred to as Higgs field are responsible for symmetry breaking between the weak, strong and electromagnetic interactions in grand unified theories. It is also connected with their potential energy density 𝑉(𝜑), which in some case may lead to the exponential (or quasiexponential, if the field 𝜑 changes slowly) expansion of the universe.
   As illustrated in Fig. 1.2, let us consider a simplest example of massive scalar fields 𝜑, whose quantum is a particle of mass 𝑚, much smaller than Planck mass. Assuming that the field is minimally coupled to gravity, its potential energy density 𝑉(𝜑) = 1/2 𝑚2𝜙2. Suppose the field 𝜑 is initially nearly homogeneous in some domain of space-time that looks locally like  an expanding universe with a growing scale factor 𝑎(𝑡) and a Hubble constant 𝐻(𝑡), defined as ȧ/𝑎. Its evolution can then be described by the usual Klein-gordon equation modified by cosmic expansion,
    (4.1)    ᾥ + 3𝐻ᾠ = -𝑚2𝜑. (from now on, ᾠ = ∂𝜑/∂𝑡 and ᾥ = ∂2𝜑/∂𝑡2)
and by the Einstein equation
    (4.2)    𝐻2 + 𝜅/𝑎2 = 4π/3𝑀𝑃2 (ᾠ2 + 𝑚2𝜑2),
where 𝜅 is +1, -1 or 0, for a closed, open or flat universe, respectively. For the closed case, one can regard 𝑎(𝑡) as the radius of the universe. The term 3𝐻ᾠ in Eq.4.1 plays the role of a friction term in the equation of motion of the field 𝜑. Eq. 4.2 tells us that if 𝜑 is initially large enough, the friction term will also large. It can be shown that if the initial field 𝜑o is greater than 1/5 𝑀𝑃, friction makes the variation of field 𝜑 very slow, so that one can neglect ᾥ in Eq. 4.1 and ᾠ in Eq. 4.2. In that case any solution rapidly approaches the regime
    (4.3)    𝜑(𝑡) = 𝜑o - 𝑚𝑀𝑃/2√(3π) 𝑡.
    (4.4)    𝑎(𝑡) = 𝑎oexp[2π/𝑀𝑃2 {𝜑o2 - 𝜑2(𝑡)}].
From these equation it follows that for times 𝑡 earlier than 𝜑o/(𝑚𝑀𝑃) the universe expanding quasiexponentially:
    (4.5)    𝑎(𝑡) = 𝑎oexp(𝐻(𝜑)𝑡)
where the Hubble constant is given by
    (4.6)    𝐻(𝜑)𝑡 ≈ √(4π/3) 𝑚𝜑/𝑀𝑃
The exponent 𝐻𝑡 is much larger than unity if 𝜑 exceeds the planck mass in these early moments. This is just the inflationary regime we wanted to obtain.
   When the field 𝜑 becomes smaller than about 1/5 𝑀𝑃, the friction term in the Eq. 4.1 becomes small, and 𝜑 oscillates rapidly near its equilibrium value of zero. If this field interacts with other matter fields, these oscillations leads to abundant particle production and heating the universe.
   The existence of an inflationary regime of this type is a general feature of a wide class of theories, including all theories in which 𝑉(𝜑) varies as 𝜑𝑛, higher-derivative theories of gravity, some versions of supergravity, and Kaluza-Klein theories. the role of the scalar field 𝜑 in these theories can be played by curvature scalar, the logarithm of the radius of compactification or the like. The remaining conditions necessary for the realization of the inflationary regime in the scenario are sufficiently natural.

5. Initial Conditions

The main idea of the chaotic inflation scenario is to consider a chaotic initial distribution of scalar field 𝜑 and to investigate its evolution. Chaotic inflation is much more general ta the new inflationary universe, in which it is assumed that the universe from the beginning was on thermodynamic equilibrium, and that 𝜑 was at the minimum of its temperature-dependent potential 𝑉(𝜑, 𝑇).
   About the chaotic initial distribution of the scalar field 𝜑, it is usually believed that for energy densities exceeding 𝑀𝑃4 quantum fluctuations of the metric are so large that one is dealing with something like a fluctuating space-time foam. The domain of initial conditions has a size greater than the 10-33 cm Plank length a density less than 𝑀𝑃4. 𝑉(𝜑) and the squares of the spatial and temporal gradients of 𝜑 are also less than 𝑀𝑃4. Also any initial distribution of the field 𝜑 inside any domain of a classical space-time is smooth enough and essentially homogeneous with in the domain.
   In case of a closed universe, when it just appears out of the space-time form or a singularity, a typical initial size should be on the order of 𝑀𝑃-1, and the initial energy density on the order of 𝑀𝑃4. Because the gradient of 𝜑 is less than 𝑀𝑃2, the initial value of the field cannot differ from some average value 𝜑0 by much more than 𝑀𝑃. The fluctuation of the energy density at the moment of creation are on the order of 𝑀𝑃4. Therefore the initial field initially "does not know" where the minimum of 𝑉(𝜑) is and later (Eq. 4.3 tells us) it rolls down to this minimum very slowly. The only constraint on the initial value of 𝜑0 is the requirement that 𝑉(𝜑) cannot exceed 𝑀𝑃4, implying that the typical initial value is of order 𝑀𝑃2/𝑚, much larger than any variation of 𝜑.
   Thus a typical initial distribution of the field 𝜑 inside a closed universe is essentially homogeneous, with a potential energy density of order 𝑀𝑃4, If the initial value of (𝛻𝜑)2 is quite reasonable factor of two or three smaller than 𝑉(𝜑) , the spatial gradient of 𝜑 very quickly becomes expontially small, and the closed universe becomes an inflationary universe by Eqs. 4.0-4.6.
   In an infinite universe, if the initial value of 𝜑 is much larger than 𝑀𝑃, then, 𝜑 remains rather homogeneous on a scale much larger than the Planck length, which is roughly the value of 𝐻-1 at that early time 𝑡. Resolving some acausal correlation between the initial values of 𝜑, the typical initial size of a classical space-time domain emerging from the space-time foam must be on the order of 𝑀𝑃-1. Outside each domain the gradient of 𝜑 is not restricted to be smaller than 𝑀𝑃2, and there is no acausal correlation between the values of the field in different domains. Only later can these domains become connected.
   In such a case the whole universe looks like a cluster of mini-universes, some of them inflationary. Just as in the closed-universe case, a typical initial value 𝜑 inside these mini-universes would  be 𝑀𝑃2/𝑚. From Eq. 4.4 we see that by the time 𝜑 approaches zero the universe has by a factor
                exp (2π𝑀𝑃2/𝑚2)
For 𝑚 on the order of 10-7𝑀𝑃, which is necessary for eventual galaxy formation, this implies that in our simplest model the inflationary domains of the universe typically expanded to 101014 times their original size!
   After a 101014-fold expansion, the geometry of space inside the expanding domain does not differ discernibly from the Euclidean geometry of flat space. Formally the term 𝜅/𝑎2 in Eq. 4.2 has become negligibly small compared with 𝐻2 after inflation.
   The observable part of the present-day universe, with the radius of order 1028 cm is a small part of the domain inflated from the smallest possible beginnings - 10-33 cm. Therefore we no longer have the horizon problem. Furthermore, after the expansion by a factor of 101014, all initial inhomogenieties, monopoles and domain boundaries have been swept beyond the horizon. This solves the primodial monopole problem and the homogeneity problem.
   The scenario which has been described is a general scheme of chaotic inflation at the level of classical physics. Surprisingly, however, quantum effects play a very important role on the inflationary scenario.

6. 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. These fluctuations can be regarded as waves of physical fields with all possible wavelengths, moving in all possible directions. If the values of these fields, averaged over some macroscopically large time, vanish, then the space filled with these fields seems to us empty and can be called the vacuum.
   In the exponentially expanding universe the vacuum structure is much more complicated according to the studies by A. Linde, A. Vilenkin, V. Mukhanov, S. Hawking, A. Starobinsky, A. Guth, P. Steinhardt, M. Turner, and others in major physics journals during the early 1980s. The wavelengths of all vacuum fluctuations of the scalar field 𝜑 grow exponentially with the expanding universe. when the wavelength of any particular fluctuation becomes grater than 𝐻-1, this fluctuation stop propagating, and its amplitude freezes at some nonzero value 𝛿𝜑(𝑥) because of the large friction term 3𝐻ᾠ in the equation of motion of field 𝜑. (𝐻-1 is ordinarily an approximation to the "age" of the universe, but during inflation it remains almost constant. In our unit system, 𝐻-1 also denotes the length 𝑐/𝐻.) The amplitude of this fluctuation then remains almost unchanged for a very long time, whereas its wavelength grows exponentially. Therefore the appearance of a classical field 𝛿𝜑(𝑥) that does not vanish after averaging over macroscopic intervals of space and time.
   Because the vacuum contains fluctuations of all wavelength, inflation leads to the creation of more and more new perturbations of the classical field with wavelength greater than 𝐻-1. The average amplitude of such perturbations generated during a time interval 𝐻-1 (in which the universe expanded by a factor of 𝑒) is given by
    (6.1)    ∣𝛿𝜑(𝑥)∣ ≈ 𝐻/2π = 𝑚𝜑/[√(3π)𝑀𝑃]
Perturbations of the field lead to density perturbations that are just right for subsequent galaxy formation if 𝑚, the mass of the quantum of 𝜑, is about 10-7𝑀𝑃. The discovery of this mechanism, which explains the origin of galaxies, was regarded in 1982 as a major success of the inflationary scenario. Now it appears that the long wave perturbations of 𝜑 are responsible not only for such local structure of the observable part of the universe, but also for the global structure of the universe.

7. Eternal Inflation

A very unusual feature of the inflationary universe is that processes separated by distances 𝑙 greater than 𝐻-1 proceed independently. Because during exponential expansion they are moving away from each other with a velocity 𝑣 exceeding the speed of light as illustrated in Fig. 1.3. 
   In this sense any inflationary domain of initial size exceeding 2𝐻-1 can be considered as a separated mini-universe, expanding independently of what occurs in the rest of the cosmos.
   Taking account of quantum fluctuations, let us consider an inflationary domain of initial size roughly 𝐻-1 containing  a sufficiently homogeneous field whose initial value 𝜑 greatly exceeds 𝑀𝑃. Eq. 4.3 tells us that  a typical time interval 𝛥𝑡 = 𝐻-1 the field inside this domain will be reduced by
    (7.1)    𝛥𝜑 = 𝑀𝑃2/4π𝜑
If 𝜑 is much less than 1/2 𝑀𝑃√(𝑀𝑃/𝑚) (about 103𝑀𝑃), this decrease 𝛥𝜑 is much larger than the amplitude of the quantum fluctuation 𝛿𝜑 generated during the same time time. But for larger 𝜑 (up to the classical limit of 107𝑀𝑃), 𝛿𝜑(𝑥)  will exceed 𝛥𝜑. Because the typical wave-length of the fluctuation field 𝛿𝜑(𝑥) generated during this time is 𝐻-1, the whole domain volume after 𝛥𝑡 will effectively have become divided into 𝑒3 separate domains (mini-universe) of diameter 𝐻-1. In almost  half of these domains the field 𝜑 grows by ∣𝛿𝜑(𝑥)∣ - 𝛥𝜑, which is not very different from ∣𝛿𝜑(𝑥)∣ or 𝐻/2π rather than decreasing, as in Fig. 1.4. During the the next time interval 𝛥𝑡 = 𝐻-1 the filed grows again in half of these mini-universe. It can be shown that the total physical volume occupied by a permanently growing field 𝜑 increases with time like 1/2 𝑒3𝐻𝑡, and the total volume occupied by a field that grows at almost same speed.
   Because the value of the Hubble constant 𝐻(𝜑) is proportional to 𝜑, the main part of the physical volume of the universe is the result of the expansion of domains with nearly the maximal possible field value, 𝑀𝑃2/𝑚, for which 𝑉(𝜑) is close to 𝑀𝑃4. There are also exponentially many domains with similar values of 𝜑. Those domains in which 𝜑 eventually becomes smaller than about 103𝑀𝑃 give rise to the mini-universe of our type. In such domains, 𝜑 eventually rolls don to the minimum of 𝑉(𝜑), and these mini-universes are subsequently describable by the usual Big Bang theory. However, the main part of the physical volume of the entire universe remains forever in the inflationary phase.
   Similar results are also valid for the old and the new inflationary scenarios, in which the main part of the volume of the universe can always remain in the state of a local extremum of 𝑉(𝜑) at 𝜑 = 0. In Linde's chaotic-inflation case, not only can the universe stay permanently on the top of the hill, it can also climb perpetually up the wall toward the largest possible values of its potential energy density, as shown in Fig. 1.5. 
   The probability for a successful climb up the wall is very small indded, but those domains in which 𝜑 jumps high enough are immediately rewarded by huge growth of their volumes. The energy source is gravitational energy with 𝑎(𝑡), the scale factor ("radius") of the universe. This gravitational energy is negative, so that the total energy of a closed universe, being the sum of the positive energy of matter and negative gravitational energy, is zero. Just this unbounded reservoir of gravitational energy makes possible the exponentially rapid growth of the total energy of matter during inflation. The negative gravitational energy seeks any opportunity to become more negative, that is, to make inflation a nonstop process.
   Thus in our scenario the universe, where there was initially at least one domain of a size on the order of 𝐻-1 filled with a sufficiently large and homogeneous field 𝜑, unceasingly reproduces itself and becomes immortal. One mini-universe produces many others, and this process goes on without end, even if some of the mini-universe eventually collapse. So our mini-universe might not be the first in the sequence illustrated in Fig. 1.6. Moreover we don't have to assume that there actually was some first mini-universe appearing from nothing or from an initial singularity at some moment 𝑡 = 0 before which there was no space-time at all.
   With the invention of the inflationary scenario one can explain why the observable part of the universe is so homogeneous and on a much larger scale the universe is extremely in-homogeneous. In some parts of the universe the energy density 𝜌 is now on the order of 𝑀𝑃4, 125 orders of the magnitude larger than the 10-29 or 10-30 g/m3. In this scenario there is no reason to assume that the universe was initially homogeneous and that all its causally disconnected parts starts their expansion simultaneously.
   If the universe is infinitely large (like the Friedmann open or flat universe), then it cannot have had a single beginning; a simultaneous creation of infinitely many causally disconnected regions is totally improbable. Therefore the universe cannot be infinite from the beginning, or it must exit eternally as a huge self-reproducing entity. Some of its parts appear at different times from singularities, or may die in a singular state. New parts are constantly being created from the space-time foam when 𝑉(𝜑) exceeds the Planck density, or they may revert to the foamlike state again as a result of large fluctuations in 𝜑. But the evolution of the universe as a whole has no end, and it may have had no beginning.

8. Mutation

In realistic theories of elementary particles there exist many different types of scalar fields 𝜑𝑖.The potential energy 𝑉(𝜑𝑖) often has many different local minima, where the universe may live for an extremely long time, much greater than 1010 years of our observable domain.
   During inflation there are large-scale fluctuations of all the fields 𝜑𝑖. As a result, the inflationary universe becomes divided into an exponentially large number of inflationary mini-universes, with the scalar fields taking all possible values. As inflation ends in some mini-universes, these scalar fields roll down to all possible minima of 𝑉(𝜑𝑖). The universe becomes divided into many different exponentially large domains, realizing all possible types of symmetry breaking between the fundamental interactions. In some of these mini-universes the low-energy physics is quite different from our own.
   It is very important that in the inflationary universe there is lots of room for all possible types of symmetry breaking and for all possible types of life. This new cosmopolitan viewpoint may greatly simplify the task of building realistic models of elementary particles. In the chaotic inflation in the higher-dimensional Kaluza-Klein theories, the energy density of the field 𝜑 grows to the Planck density, quantum fluctuations of the metric at a length 𝑀𝑃-1 become of order unity. In such domains an inflationary 𝑑-dimensional universe can sqeeze locally into a tube of smaller dimensionality 𝑑 - 𝑛 (or vice versa). If this tube is also inflationary (in 𝑑 - 𝑛 dimensions) and the initial length of tube is grater than 𝑀𝑃-1 (which is quite probable near the Planck density), then its further expansion proceeds independently of it prehistory and of he fate of its mother universe. In an eternally existing universe such processes should occur even if their probability is very small. In fact the probability of such process is small only if 𝑉(𝜑) is far below 𝑀𝑃4.
   Thus the inflationary universe becomes divided into different mini-universe where all possible types of compactification produce all sorts of dimensionalities. By this argument we find ourselves insides a four-dimensional domain with our kind of low-energy physics because our kind of life cannot exist in other domains. So there is no need to require that the results compactification and inflation have wrought in our realm be the only possible results, or the best. It is enough to find a theory in which such a compactification is possible.

9. Cosmology and Particle Physics

One can compare cosmologist and particle physicist to two groups of people tunneling a huge mountain from opposite sides. If they do not meet each other on the way through the mountain of unknown they will not have any complete theory at all.
   The inflationary-universe scenario is a spectacular manifestation of the interplay between elementary-particle theory and cosmology. At the moment, it does seem necessary to have something like inflation to obtain a consistent cosmology at peace with particle physics. The inflationary scenario is still developing rapidly.
   Just a few years ago there was no doubt that the universe was born in a single Big Bang singularity. It seemed absolutely clear that space-time was four-dimensional from the start, and that it remains four-dimensional everywhere. It was believed that if the universe is closed its total size is on  the order of its observable part, 1028 cm, and that in about 1011 years the whole universe will collapse and disappear in a Big Crunch. If, on the other hand, the universe is flat or open, then it is infinite, and it was commonly believed that its properties everywhere are approximately the same. Such a universe would exist without end, but after the decay of its protons, predicted by the unified elementary-particle theories, there would be no baryonic matter to support life of our type. The only possible choice seemed to be between the "the hot ending" in the Big Crunch and "the cold ending" in infinite empty space.
   Now it seems more likely that the universe is an eternally existing, self-producing entity, that is divided into many mini-universes much larger than our observable portion, and the laws of low-energy physics and even the dimensionality of space-time may be different in each of these mini-universes. This modification of our picture of the universe and of our place in it is one of the most important consequences of the inflationary scenario.
   In different parts of the universe different observers its endless variations. We cannot see the whole play in all its greatness, but we can try to imagine its most essential parts, and perhaps ultimately understand its meaning.

attention: some rigorous derivation might be required

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