|www.phenomenica.com (I took the image from this website)|
In http://arxiv.org/abs/1201.1298 (some more papers will follow), we study the thermodynamics of our local universe in this future. The local group is expected to collapse to a supermassive black hole that will slowly evaporate in an universe dominated by dark energy for which the simplest model is a cosmological constant. Empty space with a cosmological constant is knows as deSitter space.
We look at the dynamics between the extremely cold cosmological horizon and the hotter black hole horizon. As the black hole evaporates, heat will flow from the warmer black hole to the colder cosmological horizon producing entropy. If you miraculously add a black hole to an empty deSitter universe, you find that the presence of the black hole initially depresses the total entropy, which increases increases back to the entropy of empty deSitter as the black hole evaporates.
What is the end state of the universe? If the cosmological constant is a true constant, then the universe will reach its final thermodynamics equilibrium state in empty deSitter, i.e., empty space that expands forever. When quantum effects are included, deSitter may be unstable and decay to flat Minkowski space. Alternatively, if a slow rolling scalar field mimics the cosmological constant, this scalar field could eventually reach the bottom of its potential and then the universe would stop expanding becoming flat.
What is the entropy of flat space? Flat space would be indistinguishable from a space with a very small cosmological constant. The entropy of deSitter depends on the inverse of the cosmological constant and so as the cosmological constant gets smaller, the entropy of the universe increases. With this assumption, flat space would have a infinitely large entropy.
What about the Weyl curvature hypothesis? Perhaps you have all heard about it and wonder if it can be violated or perhaps you will hear about it now for the first time. Roger Penrose states that the difference in entropy between the initial and final state of the universe is related to the growth of the Weyl curvature. The Weyl curvature is small at the beginning of the universe, but grows with the production of singularities. However, empty deSitter space has zero Weyl curvature. We modify the conjecture by stating that the entropy does not come directly from the Weyl tensor, but from a coarse-graining over the states of the Weyl tensor. The cosmological constant limits the wavelength of the gravitational wave modes. When a black hole evaporates, the cosmological horizon grows allowing more gravitational wave modes. This increases the number of available microstates in the Weyl tensor and hence the entropy of the space. The cosmological constant is both an energy and an entropy scale. It plays a very important role in black hole thermodynamics that still has to be understood.