### 3. CLUSTERING MEASURES

**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 20*h*^{-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 8*h*^{-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).