Astronomers describe the distribution of galaxies in the universe by means of a correlation function. 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:
Given a random galaxy in a location, the correlation function describes the probability that another galaxy will be found within a given distance.
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 n-point autocorrelation functions for n greater than 2 or cross-correlation functions for particular object types are defined similarly to the two-point autocorrelation function.
The correlation function is important for theoretical models of cosmology because it provides a means of testing models which assume different things about the contents of the universe. Computer models which calculate the formation of galaxies seem to favor cold dark matter as the model with the most support.
In the spectral-analysis case they are the power spectrum and the correlationfunction; in the interferometry their counterparts are the distribution of brightness and the spatial spectrum.
Finally, equations which connect the quantities of the two levels are identical (convolution and multiplication) and include the characteristics of observations: the beam and the transfer function and their analogs, i.e., the spectral window and the correlation window.
In optics or in radio astronomy, when filled apertures are used, the maps are produced directly in the focal plane of a telescope.
Cross correlations are a useful indicator of the dependencies among different random variables as a function of time.
Correlationfunctions used in astronomy, financial analysis, quantum field theory and statistical mechanics differ only in the particular stochastic processes they are applied to with the caveat that we are dealing with "quantum distributions" in QFT.
With these definitions, the study of correlationfunctions is equivalent to the study of probability distributions.
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