In this paper we seek the best way of identifying 'objects to be' in the primordial density field, and of assigning a logical size distribution to them. We identify the objects in terms of a locally adaptive window function whose scale is determined by the neighbourhoods of the maxima. This largely takes account of the problem of hierarchical nesting of maxima and gives a mass spectrum having a form which is very similar to the classical Press-Schechter form. We present a critique of and an alternative to the standard Press- Schechter derivation of the mass function, and then go on to present an alternative view of the mass function in biased galaxy formation scenarios. Notwithstanding its apparent success, the Press-Schechter method for deriving the mass function is correct. We offer an improvement on the method that at least removes some of the major areas of doubt from their derivation. The improved result turns out to be very close to the Press- Schechter function, yet it has a quite different mathematical form. The method we propose is only strictly valid when the density contrast threshold for galaxy formation is large compared with the variance of the density fluctuations. We can remove the small-amplitude limitation by making a further departure from the Press-Schechter method and considering peaks in the density distribution having densities close to the threshold limit. This allows us to derive a 'local' window function for each incipient object and a mass function that is independent of an arbitrary choice of window function. The distribution of locally windowed peaks will of course coincide with our modification of the Press-Schechter method at high thresholds.
Monthly Notices of the Royal Astronomical Society
- Pub Date:
- July 1990