Nonperturbative Amplifications of Inhomogeneities in a Self-Reproducing Universe

TL;DR

The paper addresses nonperturbative amplification of inhomogeneities in a stationary self-reproducing inflationary universe and shows that, at a fixed time, the dominant volume for a given density $\rho$ concentrates near centers of deep wells in the density landscape, termed infloids. It develops three analytical approaches and corroborating numerical simulations to reveal that large downward quantum jumps, amplified by volume weighting, drive a stage of inflation that produces a strong, center-biased density distribution, with an amplification factor $n(\phi)$ tied to $H_{\rm max}$ and the potential slope $V'(\phi)$. While the results rely on a regularization/measure choice and thus on interpretational caveats, they provide a striking demonstration of nonperturbative effects in quantum cosmology that could have observational consequences and even challenge naïve Copernican expectations. Overall, the work illuminates how nonperturbative dynamics shape the global structure of the inflating universe and offers a framework to compare inflationary models via their volume-weighted, nonperturbative signatures.

Abstract

We investigate the distribution of energy density in a stationary self-reproducing inflationary universe. We show that the main fraction of volume of the universe in a state with a given density at any given moment of proper time t is concentrated near the centers of deep exponentially wide spherically symmetric wells in the density distribution. Since this statement is very surprising and counterintuitive, we perform our investigation by three different analytical methods to verify our conclusions, and then confirm our analytical results by computer simulations. If one assumes that we are typical observers living in the universe at a given moment of time, then our results may imply that we should live near the center of a deep and exponentially large void, which we will call infloid. Validity of this particular interpretation of our results is not quite clear since it depends on the as-yet unsolved problem of measure in quantum cosmology. Therefore at the moment we would prefer to consider our results simply as a demonstration of nontrivial properties of the hypersurface of a given time in the fractal self-reproducing universe, without making any far-reaching conclusions concerning the structure of our own part of the universe. Still we believe that our results may be of some importance since they demonstrate that nonperturbative effects in quantum cosmology, at least in principle, may have significant observational consequences, including an apparent violation of the Copernican principle.

Figures (3)

• Figure 1: Probability distribution $P(\phi)$ for $V=\lambda \phi^4/4$, $\lambda=0.1$. The dashed line is the numerical solution to a differential equation describing $P(\phi)$. The solid curve is obtained using computer simulations described in this paper. A small deviation between the solid curve and the dashed line is due to the finite size of each step and the finite grid size.
• Figure 2: Comparison between the analytical expression for $n(\phi)$ (dashed line) and the values for $n(\phi)$ obtained by computer simulations. While the analytical expression is not absolutely precise due to various assumptions (such as constancy of $g, c, s$ during the time $H^{-1}$), it does give approximately correct values for $n$.
• Figure 3: A schematic illustration which shows the number of infloids with given density and distribution of matter near their centers.