Simple explanation of Press-Schechter formalism Can someone please try to explain the Press-Schechter formalism in cosmology to me in a relatively simple manner with the most basic equations and hopefully some intuition/context? I keep getting lost in the equations and heaps of information in textbooks.
 A: General Press equation is:
$$ N(M)={\frac {1}{\sqrt {\pi }}}\left(1+{\frac {n}{3}}\right){\frac {\bar {\rho }}{M^{2}}}\left({\frac {M}{M^{*}}}\right)^{\left(3+n\right)/6}\exp \left(-\left({\frac {M}{M^{*}}}\right)^{\left(3+n\right)/3}\right) $$
Where $n$ is fluctuations power spectrum index, for scale-invariant fluctuations you can assume $n=0$, which gives a bit simpler formula. $M$ is mass of given object (galaxy, galaxy cluster or similar). And $N(M)$ is the number of a certain mass objects within a given volume of the Universe. So this relationship predicts how much of typical mass objects there will be formed due to the fluctuations power spectrum of early Universe.
As in the formula there is a very rapidly decreasing function $\exp(-M/M^*) $, it can be seen that limit :
$$ \lim_{M\to \infty} {N(M)} = 0$$, this gives an easy interpretation of formalism. The greater the mass of given object,- the less likely it to be found in the universe, and so does Press equation gives smaller amounts of more massive objects. In fact, you can state this simple truth by analyzing probability that a given star in a galaxy will be a supermassive black hole (of mass $\approx 10^6~M_☉$). Such probability would be $\approx 10^{−11}$. So supermassive black hole is not a usual guest in the galaxies. The same applies to other super-massive objects in far extreme of mass scale.
