Arrows of time and the beginning of the universe

I examine two cosmological scenarios in which the thermodynamic arrow of time points in opposite directions in the asymptotic past and future. The first scenario, suggested by Aguirre and Gratton, assumes that the two asymptotic regions are separated by a de Sitter-like bounce, with low-entropy boundary conditions imposed at the bounce. Such boundary conditions naturally arise from quantum cosmology with Hartle-Hawking wave function of the universe. The bounce hypersurface breaks de Sitter invariance and represents the beginning of the universe in this model. The second scenario, proposed by Carroll and Chen, assumes some generic initial conditions on an infinite spacelike Cauchy surface. They argue that the resulting spacetime will be nonsingular, apart from black holes that could be formed as the initial data is evolved, and will exhibit eternal inflation in both time directions. Here I show, assuming the null convergence condition, that the Cauchy surface in a nonsingular (apart from black holes) universe with two asymptotically inflating regions must necessarily be compact. I also argue that the size of the universe at the bounce between the two asymptotic regions cannot much exceed the de Sitter horizon. The spacetime structure is then very similar to that in the Aguirre-Gratton scenario and does require special boundary conditions at the bounce. If cosmological singularities are allowed, then an infinite Cauchy surface with “random” initial data will generally produce inflating regions in both time directions. These regions, however, will be surrounded by singularities and will have singularities in their past or future.

Figure 1

Arrows of time in the Aguirre-Gratton scenario with a low-entropy boundary condition (a) at the de Sitter bounce and (b) on the null hypersurface $t=-\infty$.

Figure 2

Spacetime diagram representing the Carroll-Chen scenario. The Cauchy surface is shown by a blue curved line. Black hole (and time-reversed black hole) singularities, shown by zigzag lines, are hidden behind horizons (dashed lines). The arcs represent spacelike future and past infinities of the inflating regions. High-energy inflating regions are not shown. Thermodynamic arrows of time are indicated by arrows. A past-directed null geodesic terminating at a time-reversed black hole singularity is shown by a dotted line.

Figure 3

Spacetime diagram of a singular universe with an infinite Cauchy surface. The Cauchy surface is shown by a blue curved line and the singularities by zigzag lines. The arcs represent spacelike future and past infinities of the inflating regions. Thermodynamic arrows of time are indicated by arrows.