3.1. Two-Point Correlation Functions
I have reviewed the two-point correlation function (r) earlier (section 2.1.1). The point to recall here is that the large scale structure, on scales in excess of 20h-1 Mpc., is not described by measurements of (r). There is too much noise on these scales to even say whether (r) is positive or negative. It is also worth recalling that despite what our eyes tell us, the structure on these scales is indeed linear and of small amplitude. That can be seen by smoothing the galaxy distribution with a sphere of this scale or greater. We know in fact that the variance of the optical galaxy counts averaged over spheres of 8h-1 Mpc. (ie. 800 km s-1) is unity, and this fact provides us with one basis for normalizing N-body experiments (given a bias parameter).
So why do we see all that impressive structure? The reason is that we are looking at the combined effects of large amplitude fluctuations on small scales that are correlated (albeit weakly) on large scales. The walls that define the voids are seen by virtue of their small scale structure (without which they would not look like walls!) and they look like walls or filaments because of the way they are organised on the larger scales. If we could measure the two point function reliably on the large scales we would see evidence of this.
So there is some motivation to look for clustering descriptors that quantify what our eyes tell us: that there is organised large scale structure. Note that because the structures we are seeking to quantify are linear, they may have no special dynamical significance. The Great Wall did not arise because sphere of that size collapsed to make a pancake! Not only is that dynamically unreasonable, it would also conflict with the isotropy of the microwave background radiation. Similarly, the fact that we see a great void does not imply that some region exploded from a small volume to form that space and its surrounding walls. Making that assumption would again lead to a problem with the isotropy of the microwave background (Barrow and Coles, 1990).