Correlation function (astronomy)

In astronomy, a correlation function describes the distribution of galaxies in the universe. By default, "correlation function" refers to the two-point autocorrelation function. For a given distance, the two-point autocorrelation function is a function of one variable (distance) which describes the probability that two galaxies are separated by this particular distance. It can be thought of as a lumpiness factor - the higher the value for some distance scale, the more lumpy the universe is at that distance scale.

The following definition (from Peebles 1980) is often cited:

However, it can only be correct in the statistical sense that it is averaged over a large number of galaxies chosen as the first, random galaxy. If just one random galaxy is chosen, then the definition is no longer correct, firstly because it is meaningless to talk of just one "random" galaxy, and secondly because the function will vary wildly depending on which galaxy is chosen, in contradiction with its definition as a function.

The spatial correlation function ${\displaystyle \xi (r)}$ is related to the Fourier space power spectrum of the galaxy distribution, ${\displaystyle P(k)}$, as ${\displaystyle \xi (r)={\frac {1}{2\pi ^{2}}}\int dk\,k^{2}P(k)\,{\frac {\sin(kr)}{kr}}}$

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