Number density: Initial fluctuation strength and more....

[shadishacker]

Dear all,
I am trying to understand the physics behind the "number density" formula given by Weinberg.
Is there anyone who can explain these parameters to me:

1- "Initial fluctuation strength":
\begin{equation} \rho_1= \lim_{t \to 0}\frac{\Delta \rho_M^3}{\bar{\rho}^2_M}\end{equation}
However, as t goes to 0, one can say:
\begin{equation} \rho_1= \frac{a^{-6}}{(a^{-3})^2}=1\end{equation}
in which a is the scale factor. and of course both a and the density are functions of time.
This would be because initially, we are in the mater dominated era.
So why are we using this at all?! Isn't this equal to 1?2-the time independent quantity:
\begin{equation} \tilde{\sigma}_M\end{equation}
What does it mean physically? If I want to make some plots of n, Where can find a proper function of this?3-How about:
\begin{equation} \sigma_R\end{equation}

 

[George Jones]

It would be helpful to have the context, and your reference is not very useful. Which book by Weinberg ("Gravitation and Cosmology" or "Cosmology"), and on which pages?

 

[shadishacker]

It would be helpful to have the context, and your reference is not very useful. Which book by Weinberg ("Gravitation and Cosmology" or "Cosmology"), and on which pages?

Oh sorry! you are right!
The book is Cosmology and the formula is equation (8.2.23) :
\begin{equation}
n(M,t)=\frac{\rho_1^{1/3 } (t) \bar{\rho}_M(t)}{M\sqrt{2\pi}} |\frac{d\ \tilde{\sigma}^{-1}_R (M)}{dM}|\ exp(-\frac{\rho_1^{2/3}(t)}{2 \tilde{\sigma}_{R(M)}})
\end{equation}
which shows the number density of collapsed structure with mass between M and M+dM