From the Big Bang Theory to the Theory of a Stationary Universe

TL;DR

The paper tackles the global structure of the inflationary Universe by treating the inflaton as a stochastic field whose quantum fluctuations drive diffusion and branching of inflating regions. It develops a diffusion-branching framework that yields a stationary, time-independent distribution $\tilde{P}_p(\phi,\tau|\chi)$ in sub-Planckian regimes for chaotic potentials like $V(\phi)\sim \phi^4$ and $V(\phi)\sim V_0 e^{\alpha\phi}$, with a fractal growth rate $d_{fr}=\lambda_1$. By formulating forward and backward Kolmogorov equations and implementing absorbing boundary conditions near the Planck boundary, the authors connect stochastic inflation to the Wheeler-DeWitt/Euclidean picture while highlighting the roles of the Hartle-Hawking and tunneling wave functions through stationary constructs. Computer simulations in 1D and 2D illustrate self-reproducing inflationary domains and the emergence of a stationary fractal structure, supporting a stationary cosmology where the Big Bang is no longer a singular, unique origin. The work argues for a self-consistent quantum cosmology wherein stationary states describe the long-range, global properties of the Universe, with potential observational implications for tensor-to-scalar ratios and the distribution of vacua across inflated regions.

Abstract

We consider chaotic inflation in the theories with the effective potentials phi^n and e^{αφ}. In such theories inflationary domains containing sufficiently large and homogeneous scalar field φpermanently produce new inflationary domains of a similar type. We show that under certain conditions this process of the self-reproduction of the Universe can be described by a stationary distribution of probability, which means that the 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. This represents a strong 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.

Figures (14)

• Figure 1: Time evolution of the distribution of the inflaton scalar field $\phi$, shown by the upper boundary of the shaded area, and the scalar field $\Phi$, shown by the color (black -- white). This distribution is shown as a function of one coordinate $x$, in a domain of initial size $H^{-1}(\phi_0)$. We present a sequence of three panels for each time $t$. The first one shows the distribution of the fields in the comoving coordinates, which is related to the probability distribution $P_c$. The second one exhibits the same fields in the coordinates which show the physical distance from one point to another, divided by the total distance between the two sides of the domain. The third one shows the distribution of the fields per unit physical three-dimensional volume, which would correspond to the distribution $P_p$ in a three-dimensional Universe.
• Figure 2: The distribution of the fields $\phi$ and $\Phi$ after several steps in time $\tau$.
• Figure 3: The evolution of the scalar field $\phi$ in a two-dimensional inflationary Universe in time $t$.
• Figure 4: Generation of axion domain walls during inflation.
• Figure 5: Axion domain walls in a model with a smaller value of $\Phi_0$ (Pollock Universe).