## 8. Application on the Large-scale distribution of galaxiesThe actual application of these tools is quite difficult since real
data are not easily available. To illustrate the way to proceed with
this method, we apply the above theoretical results to a practical
case with observational data, in the We will assume the validity of the assumptions made in this paper
and adopt an additional assumption here for the particular case of the
use of (
29 ), the superposition approximation, and
biasing of the
fluctuation as proportional to the where and Thus, ( 21 ) leads to: This expression resembles the one given by Peebles (1976) for , the cosmic virial theorem, but with further generality because expression ( 33 ) takes into account that . An observational was achieved by Groth & Peebles (1977): where Mpc and . The parameter is negligible for Mpc, so it is taken as zero. Also a cut-off is taken for Mpc because separated galaxies cannot be at distances less than this cut-off (if two galaxies have a distance between themselves less than this then they are considered as only one galaxy), and this ensures convergence of the integral in (33 ). In order to estimate the behaviour of , we introduce it as a power-law dependence, such that and fit the best values of
and
to solve (
33 ). The constants are derived from
observational data:
(Allen 1973),
(the actual value of the Hubble constant in
units of 100 km s If we now wish to obtain so with , and This, together with an estimate of (Allen 1973), gives us The introduction of all these data into (
33 ) and some calculations lead to the
following expression for the allowed If non-biasing, i.e. , were assumed for all scales, we would obtain the results plotted in Fig. 1, so biasing is neccesary.
The best fit, calculated numerically, is shown in Fig. 2, where left- and right-hand sides of expression ( 39 ) for parameters and Mpc are plotted. With these parameters we get or, using a different expression in the range above defined for
, where
. The obtained power law for mass correlation,
"the -2.8 power law" is different from that of objects' correlation,
"the -1.8 power law". Of course, the left and hand sides of (
39 ) were not expected to agree perfectly and
in fact they do not (Fig. 2) because we assumed a The dependence of the outcome on the cut-off value is not negligible, but neither is it very pronounced. Some numerical results were obtained with other values of the cut-off, and the qualitative result does not differ too much: a cut-off at Mpc instead of Mpc, gives us instead of 0.52, and Mpc instead of Mpc. The direct consequence of this is that mass is more correlated than
the objects (Fig. 3). This means that dark matter must exist near
galaxies and clusters of galaxies to increase the density contrast for
short distances from an object (dark matter in form of discrete unseen
galaxies, for instance dwarfs galaxies), unless the most massive
galaxies are clumped together or there is some other solution, but
something must explain the difference in both correlation functions
for galaxies and mass. Deviations from equilibrium might also be
responsible for part of the deviation attributed to biasing, as far as
it is only an approximation. I am unware of how this may affect my
results, essentially because the extent of this deviation is unknown,
though I expect it to be not very large. A large departure from
equilibrium would produce a fast evolution of the distribution and the
distributions of matter far from us would have different
distributions. This latter is not observed for long-range
correlations
The main question is how equilibrium could be reached in a short lifetime of the Universe. Violent relaxation which enormously decreases the relaxation time is a possible solution. Indeed, Saslaw (1985) points out that this must be the mechanism that governs the system due to large-scale collective modes (see chapter 38 of Saslaw 1985) and Henriksen & Widrow (1997) makes numerical simulations achieving this. The consideration of a steady state in the large-scale structure or a relaxation included in the initial conditions of the large-scale structure dynamics, before the formation of the galaxies, are other possible explanations. The deviation from equilibrium and the corrections to make to our equations to take these effects into account are topics for future papers. Further research is necessary in this area to render these results more accurate. With this example, we wanted to show the way of working with the expressions described in this paper. Readers might ask why I have not used the force equation to derive the two point-correlation function. To do this, I would need a knowledge of the biasing first, I cannot derive both things at the same time. Since the two point correlation function is better known that the bias, I decided to apply the method as described above. I had previously done some calculations to derive the two-point correlation function assuming non-biasing and the result was not compatible with observations, so I rejected this hypothesis thinking that biasing is necessary as I obtained here. © European Southern Observatory (ESO) 1997 Online publication: October 15, 1997 |