From the big bang theory to the theory of a stationary universe

We consider chaotic inflation in the theories with the effective potentials which at large $\phi$ behave either as ${\phi }^n$ or as $e^{\alpha \phi }$. In such theories inflationary domains containing a sufficiently large and homogeneous scalar field $\phi$ permanently produce new inflationary domains of a similar type. This process may occur at densities considerably smaller than the Planck density. Self-reproduction of inflationary domains is responsible for the fundamental stationarity which is present in many inflationary models: properties of the parts of the Universe formed in the process of self-reproduction do not depend on the time when this process occurs. We call this property of the inflationary Universe local stationarity. In addition to it, there may exist either a stationary distribution of probability $P_c$ to find a given field $\phi$ at a given time at a given point, or a stationary distribution of probability $P_p$ to find a given field $\phi$ at a given time in a given physical volume. If any of these distributions is stationary, we will be speaking of a global stationarity of the inflationary Universe. In all realistic inflationary models which are known to us the probability distribution $P_c$ is not stationary. On the other hand, investigation of the probability distribution $P_p$ describing a self-reproducing inflationary universe shows that the center of this distribution moves towards greater and greater $\phi$ with increasing time. It is argued, however, that the probability of inflation (and of the self-reproduction of inflationary domains) becomes strongly suppressed when the energy density of the scalar field approaches the Planck density. As a result, the probability distribution $P_p$ rapidly approaches a stationary regime, which we have found explicitly for the theories $\frac{\lambda }{4}{\phi }^4$ and $e^{\alpha \phi }$. In this regime the relative fraction of the physical volume of the Universe in a state with given properties (with given values of fields, with a given density of matter, etc.) does not depend on time, both at the stage of inflation and after it. Each of the two types of stationarity mentioned above constitutes a significant deviation of inflationary cosmology from the standard big bang paradigm. We compare our approach with other approaches to quantum cosmology, and illustrate some of the general conclusions mentioned above with the results of a computer simulation of stochastic processes in the inflationary Universe.