this is for holding javascript data
Dan Gifford edited untitled.tex
about 10 years ago
Commit id: 7a1369ced73190ea46a07ca31dd97aa7cdba7fad
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...
\begin{equation}
\langle \sigma | M \rangle = 1093 \left(\frac{h(z) M}{1e15 M_{\odot}}\right)^{0.34}
\end{equation}
This relationship has a very small lognormal scatter of
$S_{\ln(\sigma) $S_{\ln\sigma |
\ln(M)}\sim \lnM}\sim 4\%$. So:
\begin{equation}
P(\sigma | M) =
\frac{1}{\sqrt{2\pi}S_{\ln(\sigma) \frac{1}{\sqrt{2\pi}S_{\ln\sigma |
\ln(M)}} e^{\frac{(\ln(\sigma) \lnM}} e^{\frac{(\ln\sigma -
\ln(\langle \ln\langle \sigma | M
\rangle))^{2}}{2 S_{\ln(\sigma) \rangle)^{2}}{2 S_{\ln\sigma |
\ln(M)}^{2}}} \lnM}^{2}}}
\end{equation}
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 exist 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:
...
\end{equation}
Really there are completeness and purity terms in there as well, but lets ignore those for a second. So that is our expected distribution for a given $M$. The other term present is equally important $P(\hat{\sigma} | \sigma)$. This represents the probability of observing a velocity dispersion $\hat{\sigma}$ given $\sigma$. Why is this important? When we observe clusters in the real universe, we don't measure the "Evrard" velocity dispersion. We are randomly drawing from a distribution where the $\sigma$ is the expectation value. This is what \citet{Gifford13a} means by l.o.s effects. So what is that distribution? It's approximately lognormal with $S_{\log(\hat{\sigma}) | \log(\sigma)}\sim 25\%$. So:
\begin{equation}
P(\hat{\sigma} | \sigma) =
\frac{1}{\sqrt{2\pi}S_{\hat{\sigma} \frac{1}{\sqrt{2\pi}S_{\ln\hat{\sigma} |
\sigma}} e^{\frac{(\log(\hat{\sigma}) \ln\sigma}} e^{\frac{(\ln\hat{\sigma} -
\log(\sigma))^{2}}{2 S_{\hat{\sigma} \ln\sigma)^{2}}{2 S_{\ln\hat{\sigma} |
\sigma}^2}} \ln\sigma}^2}}
\end{equation}
But we are binning! That means that we have a distribution of masses in our bin that we must integrate over. What does this integral look like?