Closest mass to the line of sight

[center o bass]

Suppose we have a cosmological model where condensed objects of characteristic mass $m$ occupies some fraction $\Omega$ of the critical density $\rho_c$. Take a QSRS to be at cosmological distance $L \sim 1/H_0$ ($c=1$). Its easy to the that the average distance between the condensed masses will now be given by $\Omega \rho_c /m$.

QUESTION: why will the expectation value $\bar l$ for the distance of the closest mass to the line of sight (between us at the QSRS) be
$$ l \sim (m H_0/\Omega \rho_c)^{1/2}?$$

I've read this claim in "Method for detecting a cosmological density of condensed objects" (1973) by Press and Gunn, and would very much like to see an argument on why this is true.

 

[mfb]

I get this as geometric result. Let $\phi$ be the angle between the line of sight and directions in the sky. What is the expected number of objects N in a cone from $\phi=0$ to $\phi=\phi_0$? It is given by the volume and the density of objects: $N \propto L \frac{\Omega\rho_c}{m} \phi^2$. Set N to 1, solve for $\phi$ (and relate it to l somehow), ignore constant prefactors and you get the posted relationship.

 

[center o bass]

I get this as geometric result. Let $\phi$ be the angle between the line of sight and directions in the sky. What is the expected number of objects N in a cone from $\phi=0$ to $\phi=\phi_0$? It is given by the volume and the density of objects: $N \propto L \frac{\Omega\rho_c}{m} \phi^2$. Set N to 1, solve for $\phi$ (and relate it to l somehow), ignore constant prefactors and you get the posted relationship.

Thanks for the tip! Instead of using a cone, I used a cylinder with radius $l$ and denoted $\bar l$ by the cylinder for which there on average would be just 1 object within its volume coaxial with the line of sight!