astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...

, a correlation function describes the distribution of

galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, 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 System. ...

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, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...

. 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-oriente ...

of one

variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...

(distance); it describes the excess

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

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 Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...

(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 ...

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 In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...

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

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 fo ...

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 f ...

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