Here a dot denotes d/dt, Mp = G-1/2 is the Planck mass (using units with h = c = 1), a(t) is the cosmic scale factor, H = a/a is the Hubble parameter and k = —1,0,1 for an open, flat or closed universe, respectively. The first equation is similar to the equation of motion for a harmonic oscillator, where instead of x(t) we have <(t), so the term 3H< is like a friction effect.

If the scalar field < is initially large, the Hubble parameter H is also large from Eq. (8.1). This means that the friction term is large, so the scalar field is moving very slowly. At this stage, the energy density of the scalar field remains almost constant and the expansion of the Universe continues much faster than in the old cosmological theory. Due to the rapid growth of the scale of the Universe and slow motion of the field, soon after the beginning of this regime one has < ^ 3H<, H2 » k/a2 and <2 ^ m2<2, so the system of equations can be simplified to

The second equation shows that the scale factor in this regime grows approximately as a ~ " H = (8.4)

This stage of exponentially rapid expansion of the universe is called inflation.

When the field < becomes sufficiently small, H and the viscosity become small, inflation ends and the scalar field begins to oscillate near the minimum of V(<). As any rapidly oscillating classical field, it loses 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 Universe can be described by the standard hot big bang theory.

The main difference between inflationary theory and the old cosmology becomes clear when one calculates the size of a typical domain at the end of inflation. Even if the initial size of the inflationary Universe was as small as the Planck scale, lp ~ 10-33 cm, one can show that after 10-30 s of inflation this acquires a huge size of l ~ 1010 cm. This makes the Universe almost exactly flat and homogeneous on the large scale, because all inhomogeneities were stretched by a factor of 1010 . 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 size of the part of the Universe which we can see now, lobs ~ 10 cm. This solves most of the problems of the old cosmological theory [13].

If the Universe initially consisted of many domains, with a chaotically distributed scalar field then the domains where the scalar field was too small never inflated, so they remain small. The main contribution to the total volume of the Universe will be given by the domains which originally contained a large scalar field. Inflation of such domains creates huge homogeneous islands out of the initial chaos, each one being much greater than the size of the observable part of the Universe. That is why I call this scenario 'chaotic inflation'.

In addition to the scalar field driving inflation, realistic models of elementary particles involve many other scalar fields The final values acquired by these fields after the cosmological evolution are determined by the position of the minima of their potential energy density V(p{). In the simplest models, the potential V(p{) has only one minimum. However, in general, V(p{) may have many different minima. For example, in the simplest supersymmetric theory unifying weak, strong and electromagnetic interactions, the effective potential has dozens of different minima of equal depth with respect to the two scalar fields, $ and p. If the scalar fields fall to different minima in different parts of the Universe (a process called spontaneous symmetry-breaking), the masses of elementary particles and the laws describing their interactions will be different in these parts. Each of the parts becomes exponentially large because of inflation. In some of them, there will be no difference between weak, strong and electromagnetic interactions, and life of our type will be impossible. Other parts will be similar to the one where we live [14].

This means that, even if we are able to find the final theory of everything, we will be unable to determine uniquely properties of elementary particles; the Universe may consist of different exponentially large domains where the properties of elementary particles are 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.

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