Isotropic cosmological correlation-power-spectrum relation ($\xi (r)=\frac{1}{2{\pi }^2}{\int }_0^{\infty }{k}^{2}P(k)j_0(kr)dk$)

For the Fourier transform convention ${\delta }_{\mathbf{k}}=\int d^{3}x\delta (\mathbf{x})e^{i\mathbf{k}\cdot \mathbf{x}}$ and covariance $⟨{\delta }_{\mathbf{k}}{\delta }_{{\mathbf{k}}^{\prime }}⟩=(2\pi {)}^{3}P(k){\delta }_D(\mathbf{k}+{\mathbf{k}}^{\prime })$, inversion gives $\xi (\mathbf{r})=\int d^{3}kP(k)e^{i\mathbf{k}\cdot \mathbf{r}}/(2\pi {)}^3$. Isotropy reduces the angular integral to $4\pi j_0(kr)$, where $j_0(x)=\sin x/x$ is the order-zero Spherical Bessel function. At zero separation this gives the variance if the spectrum integral converges. A nonnegative spectrum can produce negative correlations away from the origin.