5. Results and discussion
We first discuss our analysis of the mock subsamples. Since the effect of the selection function and the geometry of the slices have both been included in generating these subsamples, our analysis of these subsamples allows us to check how well these effects are being corrected for. In the ideal situation for all the mock subsamples we should recover a flat spectrum of generalized dimensions with corresponding to a homogeneous point distribution. The actual results of the multi-fractal analysis of the mock subsamples are presented below where we separately discuss the behaviour of at small scales and at large scales .
The results for mock versions of the subsample d-12.1 are shown in Fig. 1. This is a magnitude limited subsample from a slice that has only 112 fibre fields and it contains the largest number of galaxies. We get a nearly flat spectrum with corresponding to a homogeneous point distribution at both small and large scales. Similar results are also obtained for mock versions of the other subsamples of the slice.
The analysis of mock versions of the subsample d-03.1 which contains both 112 and 50 fibre fields gives a spectrum with a weak q dependence (Fig. 2). This effect is more noticeable at small scales than at large scales. The analysis of mock versions of the d-06.1 subsample (Fig. 3) gives similar results at small scales. At large scales we get a nearly flat curve with This subsample d-06.1 has mostly 50 fibre fields and it has around half the number of galaxies as the d-12.1 subsample.
We thus find that the analysis is most effective for the subsample from the slice where shows very little q dependence and . For the other two slices we find a weak q dependence with . This clearly demonstrates that our method of multi-fractal analysis correctly takes into account the different selection effects and the complicated sampling and geometry for all the subsamples that we have considered.
We next discuss our analysis of the actual data. The analysis of the curves corresponding to versus r for the different subsamples shows the existence of two very different scaling behaviour - one at small scales and another at large scales, with the transition occurring around to . The scaling behaviour of is shown in Fig. 4 and Fig. 5 for and , respectively for the subsample d-12.1 The other subsamples all exhibit a similar behaviour. Based on this we have treated the scales (small scales) and (large scales) separately and the multi-fractal analysis has been performed separately for the small and large scales. Figs. 6, 7 and 8 show the spectrum of generalized dimensions at both small and large scales for three of the subsamples.
We find that at small scales the plots of versus q for the actual data (Figs. 6, 7, 8) are quite different from the corresponding plots for the mock versions of the data (Figs. 1, 2, 3). This clearly shows that the distribution of galaxies is not homogeneous over the scales . In addition we find that all the subsamples exhibit a multi-fractal behaviour over this range of length-scales. The interpretation of the different values of the multi-fractal dimension is complicated by the geometry of the survey and we do not attempt this here.
At large scales the behaviour of the generalized dimension is quite different. For the subsample d-12.1 the spectrum shows a weak q dependence (Fig. 6) and shows a gradual change from to as q varies from -10 to 10. This is quite different from the behaviour at small scales where the change in is larger and more abrupt. The behaviour of the other subsamples of the slice are similar. For the subsample d-03.1 we find that the spectrum is nearly flat (Fig. 7) with and for d-06.1 (Fig. 8) the spectrum is nearly flat with . These values are within the range we recover from our analysis of the mock subsamples which are constructed from an underlying random homogeneous distribution of galaxies. This agreement between the actual data and the random realizations with in all the subsamples shows that the distribution of galaxies in LCRS is consistent with a homogeneous distribution at large scales.
The work presented here contains significant improvements on the earlier work of Amendola & Palladino (1999) on two counts and these are explained below:
(1) Unlike the earlier work which has analyzed volume limited subsamples of one of the slices () of the LCRS we have analyzed both volume and magnitude limited subsamples of all the three northern slices of the LCRS. The magnitude limited samples contain more than four times the number of galaxies in the volume limited samples and they extend to higher redshifts. This allows us to make better use of the data in the LCRS to improve the statistical significance of the results and to probe scales larger than those studied in the previous analysis.
(2) We have calculated the full spectrum of generalized dimensions which has information about the nature of clustering in different environments. The integrated conditional density used by the earlier workers is equivalent to a particular point on the spectrum and it does not fully characterize the scaling properties of the distribution of galaxies.
© European Southern Observatory (ESO) 1999
Online publication: November 3, 1999