## The multiplicity function of galaxies, clusters and voids
We calculate the multiplicity function of matter condensations with given mass and defined by an arbitrary density threshold which can be a function of the mass of the objects, by directly considering the actual, deeply non-linear density field, which we compare to the popular Press-Schechter approximation. This comparison is made possible owing to an analytic description of the non-linear mass distribution, but also thanks to a modellization of the evolution of the two-body correlation function from its linear to non-linear behaviour based on Peeble's spherical collapse picture. We show the mass function is a function of a unique parameter that contains all the dependence on the mass and radius of the object. We compare this new, still simple, analytic model to a more involved formulation that should be even closer to the results obtained using the standard density threshold algorithms. This gives some hindsight into the "cloud-in-cloud" problem, for both Press-Schechter and non-linear prescriptions, and it enables us to derive the mass function of general astrophysical objects (in addition to just-virialized halos) which may be defined by any density threshold, that may even vary with the mass of the object. This is beyond the reach of usual formulations based on the initial gaussian field and gives a clear illustration of the advantages of our approach. We explain why numerical tests seem to favor both Press-Schechter and non-linear prescriptions even though the two approximations differ by their scaling as a function of mass as well as a function of redshift. We argue that numerical simulations will be closer to the non-linear predictions and should be reexamined in the light of our findings. The difference between the two is due to the fact that the Press-Schechter prescription assumes that present-day mass fluctuations can be recognized in the early linear universe, and that their number is conserved, while the non-linear approach takes into account their evolution which leads to an increase of the number of highly non-linear objects (very large or small masses). This difference is seen to be of the same magnitude as the difference obtained by varying the initial spectrum of density fluctuations. Earlier conclusions drawn about the relevance of the latter using analytical approximations to the mass function must thus be reexamined.
## Contents- 1. Introduction
- 2. The multiplicity function: non-linear approach
- 2.1. Counts-in-cells
- 2.2. Surroundings of a cell
- 2.3. Comparison with the PS approach
- 2.4. Underdense regions
- 2.5. Galaxy voids
- 3. Evolution of the correlation function
- 4. Comparison of the linear and non-linear approaches
- 5. Other astrophysical objects
- 5.1. Counts-in-cells
- 5.2. Surroundings of a cell
- 6. Conclusion
- Appendix
- Appendix A: the mass function calculated from the early gaussian fluctuations
- A.1. Gaussian approximation
- A.2. The multiplicity function: Press-Schechter formulation
- A.3. Surroundings of a cell
- Appendix B: counts in cells and matter distribution
- B.1. Gaussian limit
- B.2. Deviations from gaussian behaviour
- B.3. Properties of the density distribution as a function of smoothing scale
- B.3.1. Model-independant properties of
- B.3.2. Large scales / early times
- B.3.3. Small scales / late times
- B.4. Picture of the density fluctuations implied by hierarchical clustering
- Appendix C: non-linear models: surroundings of a cell
- C.1. general results
- C.1.1. Distribution within two volumes
- C.1.2. Limit of an infinitesimally thin corona
- C.1.3. Properties of
- C.1.4. Underdense regions
- C.2. Tree-model
- C.3. Non-constant density contrast
- Appendix D: evolution of an over(under)-density (Peebles 1980)
- Appendix E: evolution of the correlation function
- References
© European Southern Observatory (ESO) 1997 Online publication: March 26, 1998 |