= Isotropic cosmological correlation-power-spectrum relation
{title2=$\xi(r)=\dfrac1{2\pi^2}\int_0^\infty k^2P(k)j_0(kr)dk$}
For the <Fourier transform> convention $\delta_{\mathbf k}=\int d^3x\,\delta(\mathbf x)e^{i\mathbf k\cdot\mathbf x}$ and covariance $\langle\delta_{\mathbf k}\delta_{\mathbf k'}\rangle=(2\pi)^3P(k)\delta_D(\mathbf k+\mathbf k')$, inversion gives $\xi(\mathbf r)=\int d^3k\,P(k)e^{i\mathbf k\cdot\mathbf r}/(2\pi)^3$. <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.
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