String Theory and Cosmology
(2003, NS 127)
Ulf Danielsson, Ariel Goobar, Bengt Nilsson
August 14-19
Sigtuna, Sweden
Proceedings: to be published in Physica Scripta, The Royal Swedish Academy of Sciences in 2004.

 

De Sitter Space-Time Foam
by A. Starobinsky

Now, when the predictions of the simplest class of inflationary models (with one effective slowly rolling scalar field - inflaton) about the approximately flat (or, Harrison-Zeldovich-like) spectrum of initial adiabatic perturbations and the practical absence of the spatial curvature in the present Universe are convincingly confirmed by observations (note that the second prediction is actually a partial case of the first one, since an isotropic spatial curvature may be considered as a monopole perturbation of a spatially flat Friedman-Robertson-Walker background), it is just the time to return to more advanced predictions of the inflationary scenario referring to generic properties of inflationary and post-inflationary space-time for very large scales (very much larger than the present Hubble radius) where perturbations generated from quantum vacuum fluctuations during inflation are not small and their back reaction of the FRW background may not be neglected. Though these predictions are not directly observable at present, they are very important for the understanding of complexity of the whole physical World (Meta-Universe) and the place of the inflationary regime in it. The method that gives a possibility to calculate generation of perturbations in the non-linear regime and their back reaction of the locally de Sitter (before the end of inflation) or locally FRW space-time (after its end) is known as stochastic inflation. It was developed by me and other people in a number of papers beginning from 1982.

As a whole, the resulting structure of space-time during and after inflation, described in terms of many-point probability distributions for the inflaton field as a function of the logarithm of the FRW scale factor (this variable is the most convenient when speaking about space-time perturbations) or many-point correlation functions, appears to be exceedingly complicated. It is proper to call it "the de Sitter space-time foam". Generically it describes creation of infinite number of spatially disconnected post-inflationary low curvature FRW universes from one "maternal" quasi-de Sitter (inflationary) Universe. Formally, this maternal inflationary Universe, once arising, exists forever ("eternal inflation") for most inflationary models. However, for each inflationary model, there exist a finite physical correlation radius and a finite correlation number of e-folds. Thus, correlated inflation is not eternal. The same refers to the number of e-folds spent by each concrete post-inflationary Universe (in one of which we live now) at the inflationary stage. Long after this correlation e-folds number, due to loss of correlation with initial Hubble volumes, we cannot surely say that those, very distant spatially Hubble volumes originated just from this initial Hubble volume and not from another source of mechanism (the principle of the de Sitter creation "from anything"). This suggest the natural quantum interpretation of this quasi-classical de Sitter space-time foam picture as a process of continuous creation of correlated inflationary patches "from anything" which in turn either decay into multiple post-inflationary low curvature quasi-FRW universes, or (in some inflationary models) may return to the Planck curvature, to the "Planck space-time foam".