Disturbing Implications of a Cosmological Constant

In this paper we consider the implications of a cosmological constant for the evolution of the universe, under a set of assumptions motivated by the holographic and horizon complementarity principles. We discuss the “causal patch” description of spacetime required by this framework, and present some simple examples of cosmologies described this way. We argue that these assumptions inevitably lead to very deep paradoxes, which seem to require major revisions of our usual assumptions. 1 Recurrent Cosmology As emphasized by Penrose many years ago, cosmology can only make sense if the world started in a state of exceptionally low entropy. The low entropy starting point is the ultimate reason that the universe has an arrow of time, without which the second law would not make sense. However, there is no universally accepted explanation of how the universe got into such a special state. In this paper we would like to sharpen the question by making two assumptions which we feel are well motivated from observation and recent theory. Far from providing a solution to the problem, we will be led to a disturbing crisis. Present cosmological evidence points to an inflationary beginning and an accelerated de Sitter end. Most cosmologists accept these assumptions, but there are still major unresolved debates concerning them. For example, there is no consensus about initial conditions. Neither string theory nor quantum gravity provide a consistent starting point for a discussion of the initial singularity or why the entropy of the initial state is so low. High scale inflation postulates an initial de Sitter starting point with Hubble constant roughly 10−5 times the Planck mass. This implies an initial holographic entropy of about 10, which is extremely small by comparison with today’s visible entropy. Some unknown agent initially started the inflaton high up on its potential, and the rest is history. Another problem involves so-called transplanckian modes. The quantum fluctuations which seed the density perturbations at the end of inflation appear to have originated from modes of exponentially short wave length. This of course conflicts with everything we have learned about quantum gravity from string theory. The same problem occurs when studying black holes. In the naive free field theory of Hawking radiation, late photons appear to come from exponentially small wavelength transplanckian modes [1]. We now know that this is an artifact of trying to describe the complex interacting degrees of freedom of the horizon by quantum field theory defined on both the interior and exterior of the black hole. A consistent approach based on black hole complementarity [2] describes the black hole in terms of strongly interacting degrees of freedom of Planckian or string scale, and restricts attention to the portion of the space–time outside the horizon. The late time features of a universe with a cosmological constant are also not well understood. The conventional view is that the universe will end in a de Sitter phase with all matter being infinitely diluted by exponential expansion. All comoving points of space fall out of causal contact with one another. The existence of a future event horizon implies that the objects that string theory normally calculates, such as S–Matrix elements, have no