Dan Gifford edited untitled.tex  about 10 years ago

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There are several key probabilities we need to know in order to accurately predict the expectation value of some observable with mass. In more statistical language, we would like to know $P(\hat{\theta}|M)$. That is, given a cluster of mass $M$, what is the probability the observable is detected as $\hat{\theta}$. For example, the observable $\hat{\theta}$ can be velocity dispersion, some richness, or total luminosity.   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. \cite{Evrard08} \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}  \sigma = 1093 \left(\frac{h(z) M}{1e15 M_{\odot}}\right)^{0.34}  \end{equation}  This relationship has a very small scatter of $S_{\sigma | M}\sim 4\%$. The second velocity dispersion is the observed velocity dispersion $\hat{\sigma}$. In \citet{Gifford13a}, we define this as the l.o.s velocity dispersion. This has all kinds of ugly things in it including cluster shape effects, cluster environment contamination, substructure, redshift-space interlopers, and non-gaussianity. Not to mention the low number statistics that exists at low mass. Even though this is a messy observable, most are, and this is what we need to predict for a given mass $M$. So here is the generative model for observable:   \begin{equation}   P(\hat{\sigma} | M) = \sum_{\sigma} P(\hat{\sigma} | \sigma) P(\sigma | M)   \end{equation}