### Galaxy statistics calculation in Saslaw's book

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**June 21 2021**I am trying to follow a calculation from the book of William C. Saslaw, The Distribution of the Galaxies: Gravitational Clustering in Cosmology. The calculation is shown on the pages following page 122 in chapter 14 where the author talks about the Correlation function.

I am able to reproduce 14.11, which talks about the volume integral of the two-point correlation function. I understand that integrating $\xi (r)$ over the whole volume, not making a distinction on pairs of galaxies being counted twice, gives you a zero answer. I see it by solving the integral, $$\int \xi(r)d\vec{r_1}d\vec{r_2}$$ But later in the following paragraph (picture posted below),

I am unable to get the result when you do not repeat the pairing of galaxy i.e. one of the N galaxy is paired with the remaining N-1 galaxy following the same calculation that gave me the result of 14.11 (where I simply performed the integral of $\vec{r_1}$ and $\vec{r_2}$ over the entire given volume V). I am summarizing this calculation below,

$$\int \xi(r)d\vec{r_1}d\vec{r_2} = \frac{1}{\bar{n}^2}\int{<n(\vec{r_1})n(\vec{r_1}+\vec{r_2})>d\vec{r_1}d\vec{r_2}} - \int d\vec{r_1}d\vec{r_2}$$

Then you can integrate the first term under the brackets (I hope!!) and to get $$\int{n(\vec{r_1}+\vec{r_2})}d\vec{r_2}$$

Since, the integral of over $\vec{r_2}$ would just give you N (the total number of galaxies) inrespect of $\vec{r_1}$, the same from the integral of $\vec{r_1}$ it would give $N^2$. This together with the integral in the second term (which just gves you $V^2$) give you a total of 0. And as I understand, this is simply because I include infinitesimal volume $d\vec{r}$ twice when integrating. However, I am unable to do the calculation in the other case.

I would really appreciate some input from people who have done this or a similar exercise (I think it's a pretty standard calculation in galaxy statistics but I am stuck with it).

I am able to reproduce 14.11, which talks about the volume integral of the two-point correlation function. I understand that integrating $\xi (r)$ over the whole volume, not making a distinction on pairs of galaxies being counted twice, gives you a zero answer. I see it by solving the integral, $$\int \xi(r)d\vec{r_1}d\vec{r_2}$$ But later in the following paragraph (picture posted below),

I am unable to get the result when you do not repeat the pairing of galaxy i.e. one of the N galaxy is paired with the remaining N-1 galaxy following the same calculation that gave me the result of 14.11 (where I simply performed the integral of $\vec{r_1}$ and $\vec{r_2}$ over the entire given volume V). I am summarizing this calculation below,

$$\int \xi(r)d\vec{r_1}d\vec{r_2} = \frac{1}{\bar{n}^2}\int{<n(\vec{r_1})n(\vec{r_1}+\vec{r_2})>d\vec{r_1}d\vec{r_2}} - \int d\vec{r_1}d\vec{r_2}$$

Then you can integrate the first term under the brackets (I hope!!) and to get $$\int{n(\vec{r_1}+\vec{r_2})}d\vec{r_2}$$

Since, the integral of over $\vec{r_2}$ would just give you N (the total number of galaxies) inrespect of $\vec{r_1}$, the same from the integral of $\vec{r_1}$ it would give $N^2$. This together with the integral in the second term (which just gves you $V^2$) give you a total of 0. And as I understand, this is simply because I include infinitesimal volume $d\vec{r}$ twice when integrating. However, I am unable to do the calculation in the other case.

I would really appreciate some input from people who have done this or a similar exercise (I think it's a pretty standard calculation in galaxy statistics but I am stuck with it).