Cosmology and the $\mathrm{S}$ matrix

We study conditions for the existence of asymptotic observables in cosmology. With the exception of de Sitter space, the thermal properties of accelerating universes permit arbitrarily long observations, and guarantee the production of accessible states of arbitrarily large entropy. This suggests that some asymptotic observables may exist, despite the presence of an event horizon. Comparison with decelerating universes shows surprising similarities: Neither type suffers from the limitations encountered in de Sitter space, such as thermalization and boundedness of entropy. However, we argue that no realistic cosmology permits the global observations associated with an $\mathrm{S}$ matrix.

Figure 1

Conformal diagrams of Minkowski space (left) and a decelerating flat FRW universe (right). In the FRW case, any infinitesimal neighborhood of spatial infinity (circle) contains an infinite amount of matter and potentially an infinite amount of information, whereas the observer’s causal past is a finite region.

Figure 2

Conformal diagrams of flat Q-space [13, 14] (left) and de Sitter space (right). Past and future cosmological event horizons are shown. The area of the de Sitter horizon is constant, whereas the area of the Q-space horizon grows without bound at late times [13]. The Q-space initial singularity is not really null, since the curvature already becomes Planckian on a nearby spacelike slice (see Fig. 3).

Figure 3

This conformal diagram can be interpreted in three ways. It represents pure Q-space, with a spacelike singularity reflecting a Planck scale cutoff of the classical metric (see Fig. 2). It also corresponds to a big bang universe initially dominated by matter or radiation, which asymptotes to Q-space or de Sitter space at late times. The causal diamond of the observer at $r=0$ is shown. The bottom cone (B) has finite maximal area, indicating that only a finite amount of entropy enters the observable region by classical evolution. In asymptotically Q-space, however, the top cone (T) allows arbitrarily large entropy. Indeed, an unbounded number of states can be accessed by quantum fluctuations of the horizon (Sec. 4d).

Figure 4

A scalar field potential involving a false vacuum (left) gives rise to the Coleman-De Luccia solution, which describes a de Sitter region joined to an open FRW universe, in our example with vanishing vacuum energy. A conformal diagram of the expanding portion of this spacetime is shown on the right. Some examples of the orbits of the symmetry group $O(3,1)$ are shown: the domain wall, the light cone starting at $P$, and the hyperbolic time slice that includes $Q$. The region enclosed in the dotted circle is asymptotically identical with a corresponding region of the Farhi-Guth solution (Fig. 5).

Figure 5

Conformal diagram of the fully extended Farhi-Guth solution. In the text it is shown that the region marked by the dotted circle agrees asymptotically with the marked region in Fig. 4; this implies that it contains an open universe. The above diagram corresponds to the case of vanishing cosmological constant in the true minimum; for negative cosmological constant, the same argument shows that the open universe crunches.