Frustrated expectations: Defect networks and dark energy

We discuss necessary conditions for a network of cosmic domain walls to have a chance of providing the dark energy that might explain the recent acceleration of the Universe. We derive a strong bound on the curvature of the walls, which shows that viable candidate networks must be fine-tuned and nonstandard. We also discuss various requirements that any stable lattice of frustrated walls must obey. We conjecture that, even though one can build (by hand) lattices that would be stable, no such lattices will ever come out of realistic domain wall forming cosmological phase transitions. We provide some simple numerical simulations that illustrate our results and correct some misconceptions in the published literature, but a detailed numerical analysis is left for a companion paper.

Figure 1

An illustration of the decay of an unstable $X$-type junction into stable $Y$-type junctions. The decay is energetically favorable since it leads to a reduction of the total length of the walls (if all have the same tension).

Figure 2

An illustration of the collapse of three (top) and four (bottom) edge domains with $Y$-type junctions. The collapse is energetically favorable since it leads to a reduction of the total length of the walls.

Figure 3

An illustration of two different six-edged polygons with $Y$-type junctions of walls with the same tension. Both configurations have the same energy.

Figure 4

An illustration of the collapse of a four-edged polygon in the case where two of the surrounding domains are on the same vacuum state. The collapse leads to a reduction of the number of edges of contiguous domains.

Figure 5

An illustration of the collapse of a four-edged polygon in the case where all the surrounding domains are on different vacuum states and the domain walls all have the same tension. Again, the collapse leads to a reduction of the number of edges of contiguous domains.

Figure 6

An illustration of the collapse of two $Y$-type junctions into one $X$-type junction. The thickness of the traces indicates the tension strength. If the thick wall has a tension larger than twice that of the lower tension ones the collapse must happen.

Figure 7

The evolution of a perturbed square lattice with even $X$-type junctions in a matter-dominated universe. The top left panel is the initial configuration. From left to right and top to bottom panels the horizon is approximately $1/256$, $1/20$, $1/10$, and $1/5$ of the box size, respectively. The lattice stabilizes in the right bottom panel configuration.

Figure 8

The evolution of a perturbed hexagonal lattice with odd $Y$-type junctions in the matter-dominated epoch. The top left panel is the initial configuration. From left to right and top to bottom panels the horizon is approximately $1/256$, $1/10$, $1/5$, and $2/5$ of the box size, respectively. The lattice stabilizes in the right bottom panel configuration.