Do We Live in the Center of the World?

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 ρ at any given moment of time t in synchronous coordinates is concentrated near the centers of deep exponentially wide spherically symmetric holes in the density distribution. A possible interpretation of this result is that a typical observer should see himself living in the center of the world. Validity of this interpretation depends on the choice of measure in quantum cosmology. Our investigation suggests that unexpected (from the point of view of inflation) observational data, such as possible local deviations from Ω = 1, or possible dependence of the Hubble constant on the length scale, may tell us something important about quantum cosmology and particle physics at nearly Planckian densities. E-mail: [email protected] E-mail: [email protected] On leave from: Landau Institute for Theoretical Physics, Moscow. E-mail: [email protected] 1. In this paper we are going to show that according to a very wide class of inflationary theories, the main fraction of volume of the universe in a state with a given density ρ at any given moment of time t (during or after inflation) should be concentrated near the centers of deep exponentially wide spherically symmetric holes in the density distribution. Observational implications of this results depend on its interpretation. If we assume that we live in a part of the universe which is typical, and by “typical” we mean those parts of the universe which have the greatest volume with other parameters (time and density) being equal, then our result implies that we should live near the center of one of the holes in the density distribution. There should be many such holes in the universe, but each of them should be exponentially wide. Therefore an observer living near the center of any such hole will see himself “in the center of the world”. One should clearly distinguish between the validity of our result and the validity of its interpretation suggested above. Even though the effect by itself is rather surprising (its existence was first conjectured in [1]), we think that we can confirm it by several different methods. Meanwhile the validity of its interpretation is much less clear. Until the interpretation problem is resolved in one way or another, there will remain an open possibility that inflationary cosmology predicts that we should live in a center of a spherically symmetric well. As we will see, this possibility is closely related to the question whether or not Ω = 1 in inflationary universe. 2. It is well known that inflation makes the universe locally flat and homogeneous. On the other hand, during each time interval H−1 = √ 3 8πV (φ) the classical scalar field φ experiences quantum jumps of a typical amplitude δφ ∼ H 2π . (Here H is the Hubble constant; we use the system of units Mp = 1.) These jumps lead to small density perturbations with almost flat spectrum [2]. If we follow any particular inflationary domain, we can predict its behavior and calculate the typical amplitude of density perturbations in it. However, this program can be accomplished by the methods of ref. [2] only if the value of the scalar field in the initial domain is small. Meanwhile if one starts with a domain containing a sufficiently large field φ, quantum fluctuations δφ become so large that they may give rise to an eternal process of self-reproduction of new inflationary domains. For example, in the theory λ 4 φ the self-reproduction occurs in domains containing the field φ > φ∗ ∼ λ−1/6 [3]. Such a domain very soon becomes divided into exponentially many domains containing all possible values of the scalar field (i.e. all possible values of its energy density) at the same moment of time. The knowledge of initial conditions in the original domain does not allow one to predict density perturbations or even mean density inside each of these new domains. Instead of that, one may try to study those domains which may look typical (i.e. most abundant) in the context of the new cosmological paradigm. In particular, one may ask the following question. Suppose that we have one inflationary domain of initial size H−1, containing scalar field φ > φ∗. Let us wait 15 billion years (in synchronous time t in each part of this domain) and see, what are the typical properties of those