Solution of the horizon and flatness problems by multiple inflations

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Abstract
A simple treatment is developed to describe a universe that has undergone successive generic inflations of the type described by us in a previous work. By comparing general expressions for the horizon and flatness problems it is shown that allowing ‘‘failed’’ inflationary bubbles within the visible universe results in values of Ω that may depart logarithmically from unity. An exact relationship between a scale and which period of inflation determined its density perturbation amplitude is derived. Restricting attention to the case of double inflation, it is shown that many visible scales today fell within the event horizons of both de Sitter phases. It is assumed that the later inflation will set the density perturbation amplitude of such scales when they reenter the post-inflation horizon. The relative values of the amplitudes are examined and it is shown that the amplitudes created by earlier inflations will tend to be higher than that of later epochs. The relative values of the number of e-folds of the two inflations give rise to four combinations that solve the horizon problem. Treating all values as equally likely, a double-valued spectrum is obtained in only one, and not a preferred one, of those combinations.