PhD question #1: M*

In parallel with the series “Cosmological simulations” I’m starting now another series of posts about cosmology, astro/physics and related arguments. It may happen that your PhD advisor ask you a question about something and you are supposed to know the answer… but you don’t! Or it may happen that you have to pass an admittance/qualification exam to enter your PhD student career on general astrophysical knowledge but you can’t even remember some arguments exist! Because of these consideration and for my remembrance I will  write down some of these questions and I will try to answer.

These posts don’t pretend to be nor totally correct neither complete, but they reflect the answers I have found with, maybe, some corrections by other students or professors.

So let’s start with the first question: what is, how it is defined and how you can calculate $M_*$.

$M_*$ is the typical non-linear mass collapsing at the redshift we are considering. This means that $M_*$ is the typical mass of a perturbation that, at the time we are looking, has the associated liner density contrast $\delta(\mathbf{x})\sim1$, or, in the formalism of the excursion set, pass the barrier of $\delta_c=1.686$.
Starting from this and with the results of the linear theory we can obtain some qualitative laws for the non-linear evolution.
From the linear theory we have that perturbations grow in a self-similar way ($\delta(\mathbf{x},t)=\delta(\mathbf{x})D(t)$, with $D(t)$ the growth factor) and, in an Einstein-de Sitter universe (a spatially flat universe containing only matter, the Friedmann universe in which the density exactly matches the critical one), the growth factor is proportional to the scale factor.
Now, if we make a choice for the spectrum (scale-free spectrum), we can define the typical non-linear mass that is collapsing as
$$M_*(t)\propto D(t)^{6/(3+n)}$$
that, in an Einstein-de Sitter universe, becomes
$$M_*(t)\propto a^{6/(3+n)}\propto (1+z)^{-6/(3+n)}$$
where $n$ is the spectral index.
From this we can derive other relations (still in the case of a EdS universe):

• $t_*\propto (1+z)^{-3/2}$ the typical time of formation for a structure of mass $M_*$
• $\rho_*\propto (1+z)^3$
• $R\propto M_*^{1/3}(1+z)^{-1}$
• $\langle v\rangle^2M_*^{2/3}(1+z)$