350 Clustering in the universe
Figure 12.4.The typical appearance of the bidimensional correlation functionξ(rp,π),
in this specific case computed for the ESP survey [9]. Note the elongation of the contours
along theπdirection for small values ofrp, produced by high-velocity pairs in clusters.
The broken circles show contours of equal correlation in the absence of distortions.
the corresponding distribution function ofw 3 ,
f(w 3 ,r)=∫
dw 1 dw 2 F(w,r). (12.5)It is this distribution function that is convolved withξ(r)to produce the observed
ξ(rp,π). If we now callythe component of the separationralong the line of
sight, with our convention we have thatw 3 =H 0 (π−y)and the convolution
1 +ξ(rp,π)=[ 1 +ξ(r)]⊗f(w 3 ,r), (12.6)can be expressed as
1 +ξ(rp,π)=H 0∫+∞
−∞dy{ 1 +ξ[(rp^2 +y^2 )1(^2) ]}f[H 0 (π−y)]. (12.7)
Note that this expression gives essentially a model description of theeffect
produced by peculiar motions on the observed correlations, but does not take
into account the intimate relation between the mass density distribution and the
velocity field which is, in fact, a product of mass correlations (see [19] and [20]
and references therein). Within this model, therefore, we have no specific physical