370 The debate on galaxy space distribution: an overview
Assuming for simplicity a spherical sample volume with radiusRs(V(Rs)=
( 4 / 3 )πR^3 s), containingN(Rs)galaxies. The average density of the sample will
be
〈n〉=N(Rs)
V(Rs)=
3
4 πBRs−(^3 −D). (13.5)For a fractal,D<3 and its average density is a decreasing function of the sample
size:〈n〉→0forRs→∞. Then the average density depends explicitly on the
sample sizeRsand it is not a meaningful quantity.
From equation (13.2), the expression forξ(r)for a fractal distribution is [2]:
ξ(r)=(( 3 −γ)/ 3 )(r/Rs)−γ− 1. (13.6)From equation (13.6) it follows that, for the fractal sample the so-called
correlation lengthr 0 (defined asξ(r 0 )=1) is a linear function of the sample
sizeRs:
r 0 =(( 3 −γ)/ 6 )^1 /γRs. (13.7)It is then a quantitywithoutany statisticalsignificance, one simply related to the
sample size [2].
Neither isξ(r)a power law.Forr≤r 0 ,
(( 3 −γ)/ 3 )(r/Rs)−γ 1 (13.8)andξ(r)is well approximated by apower law[6].
For larger distances there is clear deviation from power-law behaviour due
to the definition ofξ(r). This deviation, however, is just due to the size of the
observational sample and does not correspond to any real change in the correlation
properties. It is clear that if one estimates the exponent ofξ(r)at distances
r ≈r 0 , one systematically obtains a higher value of the correlation exponent
due to the break ofξ(r)in the log–log plot. Only if the sample set has a crossover
to homogeneity inside the sample side, isξ(r)correct. However, this information
is given only by the(r)analysis which, for this reason, should always come
before theξ(r)investigation.
13.7 Galaxy surveys
Galaxy catalogues areangular catalogues (three-dimensional), which can be
computed inrealor inredshift space. The latter defines the galaxy positions
by theredshift distances, which is derived by thegalaxy redshift z, according to
Hubble’s law.sis not therealdistance, but contains an additional term called the
redshift distortion, which is small on scaless> 5 h−^1 Mpc [8].
We will report the statistical properties ofredshift surveys, which contain the
large majority of avalaible three-dimensional data.