# Deducing distance of closest neighbor galaxy from expression of correlation and mean galaxy density

The original post was on physics exchange but I prefer to move it here :

I am trying to estimate the distance of closest galaxy neighbor knowing the expression of number of neighbors $$\text{d}N$$ into a volume $$\text{d}V$$, the mean density $$n_\text{gal}$$ and the correlation function, i.e with this expression:

$$\text{d}N=n_{\text{gal}}\,\text{d}V\,(1+\xi(r))$$

with $$\xi(r)=\bigg(\dfrac{r}{r_{0}}\bigg)^{-\gamma}\quad\text{with}\quad\gamma\,\sim\,1.77 \quad\text{and}\,r_{0}\,\sim\,5\,\text{Mpc}$$.

I must precise that expression of $$\xi(r)$$ is valid for $$r$$ between $$0.5\,\text{and}\,10\,\text{Mpc}$$

I take into my calculation the following value for $$n_{\text{gal}}=0.0420\,\text{h}^{-3}\,\text{Mpc}^{-3}$$ : I tried to choose a typical value for this density but it depends on which scale we consider (galaxy cluster, super cluster, very large scales ...), so maybe don't focus on this value.

What it interests me is to have elements to answer to the following question : How could I infer the distance of the closest galaxy from us?

UPDATE 1: Maybe I should take the lower limit of validity for the definition of $$\xi(r)$$, i.e $$r=0.5\,\text{Mpc}$$ and so:

$$\text{d}N/\text{d}V=n_{\text{gal}}\,(1+\xi(r=0.5))$$

So I would have:

$$\text{D}_{\text{closest}}=\bigg(n_{\text{gal}}(1+(0.5/5)^{-1.77})\bigg)^{-1/3}=0.7353\,\text{Mpc}$$

But by taking $$r=0.5\,\text{Mpc}$$, I am already located myself like the closest neighbor was at $$r=0.5\,\text{Mpc}$$, am I not ?

Is this reasoning correct?

Any help is welcome

• Yes I would say is a circle. ... – Alchimista Dec 23 '18 at 8:57
• @Alchimista . Do you think this is the right method ? As I said, But by taking $r=0.5\,\text{Mpc}$, I am already located myself like the closest neighbor was at $r=0.5\,\text{Mpc}$, am I not ? this would be an issue, no ? – youpilat13 Dec 23 '18 at 15:11
• No I am saying that it seems that you impose the result to be that.... I have the same doubt that you have. – Alchimista Dec 24 '18 at 8:01
• But it somehow correct that if there is a parameter that holds between, say 0.5 and 10, something should be found int between. But indeed one could have chosen every value in that range perhaps. I don't know much about correlation. I would naively work without it once I know the total number of galaxies in a sphere of radius 10 Mpc. – Alchimista Dec 24 '18 at 8:09
• Sorry that I can only comment. In my mind the corf funct account for dishomogeneity at various scales. Once you identify a range you know the average number of objects around. From that you consider them homogeneously distributed in a sphere (or a cube for that it matters) and infer the distance between two neighbours. One is you and the second the closest galaxy. – Alchimista Dec 24 '18 at 8:36