Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 4 / a / Solution

Let $\delta (\mathbf{x})$ be a mean-zero density contrast. For a statistically homogeneous and isotropic field, its two-point correlation function is $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$, depending only on $r=|\mathbf{r}|$. For a galaxy point process the corresponding distinct-pair definition is $d{P}_{12}={\bar{n}}^2[1+\xi (r)]d{V}_{1}d{V}_2$, the excess pair probability relative to independent uniform positions. Self-pair shot noise is separated when using the density-field version.

Specify the Fourier convention byThe diagonal covariance follows from statistical homogeneity, and isotropy makes the matter power spectrum a function of $k$ alone. Substitute the inverse transforms into the correlation and integrate the delta function:Choose the polar axis parallel to $\mathbf{r}$. Angular integration gives $\int d{\Omega }_k{e}^{ikr\cos \theta }=4\pi \sin (kr)/(kr)$. HenceConversely, when the transform is defined, $P(k)=4\pi {\int }_0^{\infty }{r}^2\xi (r)\sin (kr)/(kr)dr$. The stated convention fixes every $2\pi$ factor.