deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index 722acb6..b0b2931 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
...
Let's take velocity dispersion as an example here. There are two relevant velocity dispersions in our studies. The first is the true 3D/2D velocity dispersion $\sigma$ which we cannot measure in the real universe. \citet{Evrard08} measured $\sigma$ for halos in N-body simulations and showed that they relate to the critical mass $M_{200}$ of the host halo on a very tight relation  \begin{equation}  \langle \sigma | M \rangle = 1093 \left(\frac{h(z) M}{1e15 M_{\odot}}\right)^{0.34} M}{1e15}\right)^{0.34}  \end{equation}  This relationship has a very small lognormal scatter of $S_{\ln\sigma | \ln M}\sim 4\%$. So:  \begin{equation}