3.3. Fractal scaling
The number of neighbors - on average - a galaxy has within a sphere of radius r is just the integral
When this function follows a power-law N( < r) rD2, the exponent is the correlation dimension, and the point pattern is said to verify fractal scaling. There is no doubt that up to a given scale the galaxy distribution fits rather well the fractal picture. However, some controversy regarding the extent of the fractal regime has motivated an interesting debate [26, 27, 28, 7]. Nevertheless, the new data is showing overwhelming evidence that the correlation dimension is a scale dependent quantity. Different authors [29, 30, 31, 32, 33, 34, 35, 36, 37] have analyzed the more recent available redshift surveys using N(r) or related measures with appropriate estimators. Their results show unambiguously an increasing trend of D2 with the scale from values of D2 ~ 2 at intermediate scales to values of D2 ~ 3 at scales larger than 30 h-1 Mpc. Moreover, one of the strong predictions of the fractal hypothesis is that the correlation length - the value r0 at which the correlation function reaches the unity ((r0) = 1) - must increase linearly with the depth of the sample. This seems not to be the case [38, 39].