Inflationary Cosmology

Inflationary Cosmology

 Inflationary Cosmology
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According to the big bang theory, the rate of expansion of the universe given by the Hubble “constant” H is large when the density of the universe is large. If the universe is filled by ordinary matter, then its density rapidly decreases as the universe expands. Therefore expansion of the universe rapidly slows down as its density decreases. This rapid decrease of the rate of the universe expansion is the main reason of all our problems with the standard big bang theory. However, because of the equivalence of mass and energy established by Einstein (E = mc2), the potential energy density V() of the scalar field  also contributes to the rate of expansion of the universe. In certain cases the energy density V() decreases much more slowly than the density of ordinary matter. This may lead to existence of a stage of extremely rapid expansion (inflation) of the universe.

There are several different versions of inflationary theory. Let us consider the simplest model, which I called chaotic inflation. This model describes a scalar field  with a mass m and with the potential energy density V() = m22 /2.Since energy density has a minimum at  = 0, one may expect that the scalar field  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.

Now we have only one step to make in order to understand where inflation comes from. If the scalar field  initially was large, its energy density V() was also large, and the universe expanded very rapidly. Because of this rapid expansion the scalar field was moving to the minimum of V() very slowly, as a ball in a viscous liquid. Therefore at this stage the energy density V(), unlike the density of ordinary matter, remained almost constant, and expansion of the universe continued with a much greater speed than in the old cosmological theory: The size of the universe in this regime grows approximately as eHt, where H is the Hubble constant

To understand the situation at a more formal level one should analyze two equations which describe inflation in our model: d2/d2t +3H d/dt = -V’(), and H2 =8GV()/3. The second equation is a slightly simplified Einstein equation for the scale factor (radius) of the universe a(t); H is a Hubble constant, H=(da/dt)/a, G is the gravitational constant. The term 3H d/dt in the first equation is similar to the friction (viscosity) term in the equation of motion for a harmonic oscillator. One can show that if V() is approximately constant during a sufficiently long period of time, the last equation has an inflationary solution a(t)  eHt.

This stage of self-sustained exponentially rapid expansion of the universe was not very long. In a realistic version of our model its duration could be as short as 10-35 seconds. When the energy density of the field  becomes sufficiently small, viscosity becomes small, inflation ends, and the scalar field  begins to oscillate near the minimum of V(). 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 question shows that even if the size of the part of inflationary universe at the beginning of inflation in our model was as small as lP = 10-33 cm, after 10-35 seconds of inflation this domain acquires a huge size l  101000000000000 cm! These numbers are model-dependent, but in all realistic models this size appears to be many orders of magnitude greater than the size of the part of the universe which we can see now, l  1028 cm. This immediately solves most of the problems of the old cosmological theory.

Our universe is so homogeneous because all inhomogeneities were stretched 101000000000000 times. The density of primordial monopoles becomes exponentially diluted by inflation. The universe becomes enormously large. Even if it was a closed universe of a size  10-33 cm, after inflation the distance between its “South” and “North” poles becomes many orders of magnitude greater than 1028 cm. We see only a tiny part of the huge cosmic balloon. That is why the universe looks so flat. That is why nobody has ever seen how parallel lines cross. That is why we do not need expansion of the universe to begin simultaneously in 1088 different causally disconnected domains of a Planck size. One such domain is enough to produce everything, which we can see now!

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