astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...

, a correlation function describes the distribution of objects (often

stars A star is a luminous spheroid of plasma held together by self-gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night; their immense distances from Earth make them appear as fixed points of ...

galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, and dark matter bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar Sys ...

) in the universe. By default, "correlation function" refers to the two-point

autocorrelation function Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...

. The two-point autocorrelation function is a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

of one variable (distance); it describes the excess

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

of finding two galaxies separated by this distance (excess over and above the probability that would arise if the galaxies were simply scattered independently and with uniform probability). It can be thought of as a "clumpiness" factor - the higher the value for some distance scale, the more "clumpy" 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

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

. Assuming the universe is

isotropic In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...

(which observations suggest), the correlation function is a function of a

scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...

distance. The two-point correlation function can then be written as

\xi_2(\left,  \mathbf x_1 - \mathbf x_2 \) = \langle \delta(\mathbf x_1) \delta(\mathbf x_2) \rangle,

where

\delta(\mathbf x) = (\rho(\mathbf x) - \bar\rho)/\bar\rho

is a unitless measure of overdensity, defined at every point. Letting

\Delta = \left,  \mathbf x_1 - \mathbf x_2 \

, it can also be expressed as the integral

\xi_2(\Delta) = \frac \int d^3 x \, \delta(\mathbf x) \delta(\mathbf x + \mathbf \Delta) .

The spatial correlation function

\xi(r)

is related to the Fourier space

power spectrum In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...

of the galaxy distribution,

P(k)

, as

\xi(r) = \frac \int dk \, k^2 P(k) \, \frac{kr}

The ''n''-point autocorrelation functions for ''n'' greater than 2 or

cross-correlation In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a ''sliding dot product'' or ''sliding inner-product''. It is commonly used f ...

functions for particular object types are defined similarly to the two-point autocorrelation function. The correlation function is important for theoretical models of

physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...

because it provides a means of testing models which assume different things about the contents of the universe.

References

Peebles, P.J.E. 1980, The large scale structure of the universe

Theuns, Physical Cosmology
Extragalactic astronomy Covariance and correlation

See also

References