8. Application on the Large-scale distribution of galaxies
The 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 N -body system of the large scale distribution of galaxies in the Universe (see for example Peebles 1980; Borgani 1995) which is a homogeneous and isotropic distribution. A recent model, using equilibrium statistical mechanics as well as other considerations, was also developed in Pérez-Mercader et al. (1996).
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 n fluctuation where the constant of proportionality depends on the scale:
where b is the biasing parameter, dependent a priori on r, the distance between both particles. Astronomers usually make a stronger assumption by taking b to be constant, but I am not going to be so restrictive. I adopt a unit system where and set . As a consequence,
Thus, ( 21 ) leads to:
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-1 Mpc-1 ) and km s-1 (Allen 1973).
If we now wish to obtain C, according to ( 24 ), we must convert G from our units (time unit: second; length unit: 1 Mpc; mass unit: the average mass of a galaxy, in kg) to the MKS system, multiplying by a factor of . Thus, denoting by the value of G in the MKS system, we have
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 r values:
If non-biasing, i.e. , were assumed for all scales, we would obtain the results plotted in Fig. 1, so biasing is neccesary.
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 b dependence, a power law, which might not be very realistic. A worthwhile result is that a deviation from is necessary to get an agreement between the left- and right-hand sides of ( 39 ), and the solution must be close to . This result increases the information about the mass-mass correlation function (see Peebles 1980), which was unknown till now. If this result is true and is verified by alternative methods, it would give us important information about the "dynamics" in the large scale distribution of matter in the Universe.
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 correlations1 , so we must infer that the departure from equilibrium conditions cannot be too high unless perhaps for very small ranges. For typical scales in galaxy clusters, we also find strong evidence of equilibrium (Carlberg et al. 1997). This discussion is beyond the scope of this paper. In Saslaw & Hamilton (1984, their Sect. 6), we find further arguments in favour of this: "Gravitational N -body experiments give good agreement with the theory. This shows that even though an equilibrium theory may not explain the ultimate fate of galaxy clustering, it does provide a good description over extremely long time scales when the correlations are `frozen out' by the expansion of the Universe...[T]he equilibrium theory seems to explain most of the results."
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