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Let the distribution of galaxies be a number like $n$. we want to find the total number of galaxies within the radius $r$ from a specified point. in which shape of the universe (flat, open, closed) the number we get is higher? ($n,r$ are the same in this universes and only the area (volume) in each universe is different).

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  • $\begingroup$ By distribution do you mean something like the number density? I don't see how you can represent "the distribution of galaxies" by a single number otherwise. $\endgroup$ – zephyr Jul 20 '17 at 18:51
  • $\begingroup$ @zephyr . yes I mean number density. I want to find for example for r = 500 MPc, In which type of universe the total number of galaxies are higher? $\endgroup$ – titansarus Jul 20 '17 at 20:48
  • $\begingroup$ As a hint, do you have equations that allow you to calculate distances in the universe that depend on the curvature? Usually it will depend on either $k$ or $\Omega_k$. $\endgroup$ – zephyr Jul 20 '17 at 20:55
  • $\begingroup$ So what else changes between your three universes? Or are we allowed to assign arbitrary values for $\Omega_m$, $\Omega_\lambda$, $H_0$ etc.? I'm a bit rusty with the Friedmann equations, but I'm quite sure you need to provide some more information for a meaningful answer. $\endgroup$ – Alex Jul 20 '17 at 21:15
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It seems clear to me that this is a homework question. I believe what the question is after is to look at the equations for distances in the universe as a function of curvature and see how this curvature affects things. Read, for example, sources like this one or this one.

As with most of things in cosmology, there's lots of equations and many ways to write them. A relevant equation would be the following angular diameter distance:

$$D_M = (1+z)^{-1} \begin{cases} \frac{1}{\sqrt{\Omega_k}}\sinh\left(\sqrt{\Omega_k}D_c/D_H\right), & k = -1 \\ D_c, & k = 0 \\ \frac{1}{\sqrt{\left|\Omega_k\right|}}\sin\left(\sqrt{\left|\Omega_k\right|}D_c/D_H\right), & k = +1 \\ \end{cases} $$

In this case, you can see the three forms it takes for a closed universe ($k=-1$), a flat universe ($k=0$), and an open universe ($k=+1$). It should be pretty clear that if you chose some volume with some radius calculated by $D_A$, you'd get different volume sizes and thus different amounts of galaxies in each volume for the three universe types.

By looking at the equations and playing around with them, you should be able to convince yourself which universe type results in the largest volume.

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