C. Understanding structure
We have tried and tested a number of descriptors of the galaxy distribution with varying amounts of success. The task has been helped by ever-growing data sets, but it is nevertheless becoming clear that a somewhat different approach may be required if we are to improve substantially on what we understand now.
What different approaches might we take? Our visual impression of large scale structure is that it is dominated by voids, filaments and clusters. This suggests that instead of looking at sample-wide statistical measures such as correlation functions, we might try to isolate the very features that strike us visually and examine them as individual structures. Much effort has already been devoted to isolating "clusters" of galaxies, but there are currently few, if any, methods available for isolating either voids or filaments.
Wavelet analysis and its generalizations such as Beamlets and Ridgelets may prove useful in identifying these structures (Donoho et al., 2002). Other nonlinear analysis methodologies exist but have not been tried in this context. The fact that galaxies (or points in a simulation) provide a sparse Poisson sample of the underlying data complicates the application of what might otherwise be standard methods.
The power of having a clear mathematical descriptor lies in being able to unambiguously identify and study specific objects. This in turn provides a tools for confronting simulations with data.