 Solution

For a statistically homogeneous matter distribution, the cosmological two-point correlation function of the mean-zero density contrast $\delta =(\rho -\bar{\rho })/\bar{\rho }$ isSpatial homogeneity removes dependence on $\mathbf{x}$, and isotropy reduces the separation argument to its magnitude. For discrete galaxies, the galaxy two-point correlation function equivalently specifies the excess pair probability at distinct positions:Consequently the mean density of other galaxies, conditioned on a galaxy at the origin, is $\bar{n}[1+\xi (r)]$. The self-pair at zero separation is excluded from this definition; its Dirac delta contribution is the galaxy power spectrum shot noise of the unsmoothed number-density field.