<p>The point of the two-point correlation function (pun not intended) is to describe how clustered the galaxies in the universe are. Astronomers want to know if they're all bunched up in tight bundles with huge voids, or maybe they're more or less uniformly distributed.</p>
<p>The two-point correlation function is a way of measuring the distribution of these galaxies <em>compared to a random distribution</em>. In other words, if your distribution of galaxies is not uniformly random (which it isn't for our universe - our universe has a distinct non-random structure of clusters and voids) then your correlation function will indicate the degree to which it isn't random. Let's take a look at the last equation listed in your link.</p>
<blockquote>
<p>The most commonly-used estimator is from <a href="http://cdsads.u-strasbg.fr/abs/1993ApJ...412...64L" rel="nofollow noreferrer">Landy &amp; Szalay (1993)</a></p>
<p><span class="math-container">$$\xi(r) = \frac{1}{RR(r)}\Big[DD(r)\Big(\frac{n_R}{n_D}\Big)^2 - 2DR(r)\Big(\frac{n_R}{n_D}\Big)+RR(r)\Big]$$</span></p>
</blockquote>
<p>The symbols in this equation are defined as follows:</p>
<ul>
<li><span class="math-container">$\xi(r)$</span> - The two-point correlation function.</li>
<li><span class="math-container">$r$</span> - The separation between two galaxies.</li>
<li><span class="math-container">$DD$</span>,<span class="math-container">$DR$</span>,<span class="math-container">$RR$</span> - The counds of pairs of galaxies as a function of separation in the data catalogue, between the data and random catalogue, and in the random catalogue, respectively.</li>
<li><span class="math-container">$n_D$</span>, <span class="math-container">$n_R$</span> - The mean number density of galaxies in the data and random catalogues, respectively.</li>
</ul>
<p>If your data catalogue happens to be completely random, then <span class="math-container">$DD=DR=RR$</span> and you find that <span class="math-container">$\xi(r)=0$</span> (assuming also that <span class="math-container">$n_D=n_R$</span>), showing your data catalogue is indeed indistinguishable from a random distribution. If, however, your data catalogue is not defined with completely random galaxies, you'll find <span class="math-container">$\xi(r)\ne0$</span>, and the degree to which it is non-zero indicates how non-random, or organized your system is.</p>

CC BY-SA 3.0
2016-10-01T21:12:04.620Z
To find out if the clustering is not random, we can simply do so by calculating the mean density of the data catalogue within circles of a given radius at different points as the centre. 
Why do we need to create something called as a random catalogue?
Srivatsan
CC BY-SA 3.0
2016-10-01T22:38:53.157Z
@ThePredator I don't see how calculating the mean density as a function of radius from some (arbitrary) point tells you anything about the nature of the distribution. You need a random catalogue so you can compare your catalogue to that random catalogue and determine how different it is. How can you compare A to B if you have no B to compare to?
zephyr
CC BY-SA 3.0
2016-10-02T07:40:02.363Z
Since clusters of galaxies are not random, if we are going to calculate the density at several points as the centre inside the catalogue, we will be finding different values at different points. Doesn't this tell us that it is different from a random catalogue (as in a random catalogue we will find the density to be almost even at any point).
Srivatsan
CC BY-SA 3.0
2016-10-03T12:38:22.680Z
@ThePredator If I understand you correctly, you want to calculate the radial density from several points to compare them. This won't show that it isn't random though. This will only show that your catalogue is homogeneous, i.e., the same everywhere. But it could be random and the same, or clustered and the same. You have to compare your catalogue to a random one to determine if it is random. Comparing it to itself tells you nothing about the clustering.
zephyr
CC BY-SA 3.0
2016-09-30T19:32:18.793Z
2016-09-30T19:32:18.793Z
2020-06-17T09:47:40.653Z
zephyr
<p>To study clustering of galaxies and clusters, one makes use of
<a href="https://ned.ipac.caltech.edu/level5/March12/Coil/Coil2.html" rel="nofollow">the two-point correlation function</a>. However, we can see that all 
the estimators of $\xi(r)$ use a random catalogue. </p>

<p>What is the random catalogue used for? </p>

<galaxy><galaxy-cluster>
What is the role of the random catalogue in calculating the two-point correlation function?
Srivatsan
# What is the role of the random catalogue in calculating the two-point correlation function?

(Asked by: username_0 on 2016-09-30)

To study clustering of galaxies and clusters, one makes use of
[the two-point correlation function](https://ned.ipac.caltech.edu/level5/March12/Coil/Coil2.html). However, we can see that all
the estimators of $\xi(r)$ use a random catalogue.

What is the random catalogue used for?

**Tags:** galaxy, galaxy-cluster

---

## 1 answers

### Answer by username_1 on 2016-09-30. Score: 3 | Accepted answer

The point of the two-point correlation function (pun not intended) is to describe how clustered the galaxies in the universe are. Astronomers want to know if they're all bunched up in tight bundles with huge voids, or maybe they're more or less uniformly distributed.

The two-point correlation function is a way of measuring the distribution of these galaxies *compared to a random distribution*. In other words, if your distribution of galaxies is not uniformly random (which it isn't for our universe - our universe has a distinct non-random structure of clusters and voids) then your correlation function will indicate the degree to which it isn't random. Let's take a look at the last equation listed in your link.

> The most commonly-used estimator is from [Landy & Szalay (1993)](http://cdsads.u-strasbg.fr/abs/1993ApJ...412...64L)
>
> $$\xi(r) = \frac{1}{RR(r)}\Big[DD(r)\Big(\frac{n\_R}{n\_D}\Big)^2 - 2DR(r)\Big(\frac{n\_R}{n\_D}\Big)+RR(r)\Big]$$

The symbols in this equation are defined as follows:

* $\xi(r)$ - The two-point correlation function.
* $r$ - The separation between two galaxies.
* $DD$,$DR$,$RR$ - The counds of pairs of galaxies as a function of separation in the data catalogue, between the data and random catalogue, and in the random catalogue, respectively.
* $n\_D$, $n\_R$ - The mean number density of galaxies in the data and random catalogues, respectively.

If your data catalogue happens to be completely random, then $DD=DR=RR$ and you find that $\xi(r)=0$ (assuming also that $n\_D=n\_R$), showing your data catalogue is indeed indistinguishable from a random distribution. If, however, your data catalogue is not defined with completely random galaxies, you'll find $\xi(r)\ne0$, and the degree to which it is non-zero indicates how non-random, or organized your system is.

> **Comments:**
> **username_0**: To find out if the clustering is not random, we can simply do so by calculating the mean density of the data catalogue within circles of a given radius at different points as the centre.  Why do we need to create something called as a random catalogue?  
> **username_1**: @ThePredator I don't see how calculating the mean density as a function of radius from some (arbitrary) point tells you anything about the nature of the distribution. You need a random catalogue so you can compare your catalogue to that random catalogue and determine how different it is. How can you compare A to B if you have no B to compare to?  
> **username_0**: Since clusters of galaxies are not random, if we are going to calculate the density at several points as the centre inside the catalogue, we will be finding different values at different points. Doesn't this tell us that it is different from a random catalogue (as in a random catalogue we will find the density to be almost even at any point).  
> **username_1**: @ThePredator If I understand you correctly, you want to calculate the radial density from several points to compare them. This won't show that it isn't random though. This will only show that your catalogue is homogeneous, i.e., the same everywhere. But it could be random and the same, or clustered and the same. You have to compare your catalogue to a random one to determine if it is random. Comparing it to itself tells you nothing about the clustering.  

---

