Now it as an interesting question whether single stream regions are percolating. That means: is there an infinetly large single stream region spanning the whole universe, or do they form many small independent regions? With the simulation we present here, we tried to test this numerically. We find that the connectivity in Eulerian space shows a strong resolution dependence, but the more robust test in Lagrangian space shows no percolation.
Linking in Eulerian Space
We find that even at relatively high resolutions, the size of the largest single stream regions still depends on the resolution of the single stream field. Below we show a slice-video that compares three different resolutions with each other. The top left uses a mesh resolution of 512^3 bins corresponding to a cell width of 78kpc, the bottom left side uses 1024^3 bins (cell width 39 kpc) and the bottom right 2048^3 bins (20kpc cell width). The top right shows the corresponding density field. The higher resolution cases tend to be far more connected than the lower resolutions. That means that large regions get chained together through gaps that are smaller than 39 kpc.
From the movie above it looks like the 2048^3 bin case percolates in all three dimensions. However, the region is not connected to itself in the x- and z-direction, but it only percolates in the y-dimension. That means with its periodic replicants it would form an infinite string like structure in the vertical direction. We show below a movie where it becomes more clear where the region is disconnected: the left side shows the periodically connected regions from before, but the right side shows the connected components when ignoring the periodic boundary conditions. The top shows the 1024^3 bin case whereas the bottom shows the 2048^3 bins case. In many cases the left side shows neighbouring regions with the same color which are however not connected as the right side suggests.
While the linking of single stream regions in Eulerian Space seem to be affected by numerical artefacts, we think that linking the single stream regions in Lagrangian Space should be more robust. This is completely equivalent in the continuum limit. However, at limited resolution there are two advantages: In Lagrangian space (1) the surface area of the single-stream - multi-stream intersection is far smaller and (2) the disconnecting multi stream regions are much thicker. Therefore it is far more unlikely to link regions together through unphysical gaps in Lagrangian space than in Eulerian space.
Below we show on the left side the labels from the 2048^3 bin case mapped into Lagrangian space. The colors are the same as the movie above in the bottom left panel. On the right side we show the regions that we find to be connected if we apply the linking on the single stream mesh in Lagrangian space instead of Eulerian space. Several of the regions which appear to be connected in the Eulerian linking are actually disconnected in Lagrangian space. This likely means that the Eulerian version connects in these cases through numerical artefacts. The Lagrangian linking does not show percolation.
