In astronomy, a correlation function describes the distribution of galaxies in the universe. By default, "correlation function" refers to the two-point autocorrelation function. The two-point autocorrelation function is a function of one variable (distance); it describes the excess probability 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.
From every pair in a distribution of galaxies, the two-point correlation function is calculated by counting the number of pairs that are separated by distances in various bins.
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.
Assuming the universe is isotropic (which observations suggest), the correlation function is a function of a scalar distance. The two-point correlation function can then be written as
where is a unitless measure of overdensity, defined at every point. Letting , it can also be expressed as the integral
The spatial correlation function is related to the Fourier spacepower spectrum of the galaxy distribution, , as
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 physical cosmology because it provides a means of testing models which assume different things about the contents of the universe.
Peebles, P.J.E. 1980, The large scale structure of the universe
Theuns, Physical Cosmology
December 15, 2023
correlation, function, astronomy, other, uses, correlation, function, disambiguation, astronomy, correlation, function, describes, distribution, galaxies, universe, default, correlation, function, refers, point, autocorrelation, function, point, autocorrelatio. For other uses see Correlation function disambiguation In astronomy a correlation function describes the distribution of galaxies in the universe By default correlation function refers to the two point autocorrelation function The two point autocorrelation function is a function of one variable distance it describes the excess probability 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 source source source source source source source source From every pair in a distribution of galaxies the two point correlation function is calculated by counting the number of pairs that are separated by distances in various bins 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 Assuming the universe is isotropic which observations suggest the correlation function is a function of a scalar distance The two point correlation function can then be written as3 2 x 1 x 2 d x 1 d x 2 displaystyle xi 2 left mathbf x 1 mathbf x 2 right langle delta mathbf x 1 delta mathbf x 2 rangle where d x r x r r displaystyle delta mathbf x rho mathbf x bar rho bar rho is a unitless measure of overdensity defined at every point Letting D x 1 x 2 displaystyle Delta left mathbf x 1 mathbf x 2 right it can also be expressed as the integral 3 2 D 1 V d 3 x d x d x D displaystyle xi 2 Delta frac 1 V int d 3 x delta mathbf x delta mathbf x mathbf Delta The spatial correlation function 3 r displaystyle xi r is related to the Fourier space power spectrum of the galaxy distribution P k displaystyle P k as3 r 1 2 p 2 d k k 2 P k sin k r k r displaystyle xi r frac 1 2 pi 2 int dk k 2 P k frac sin kr kr 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 physical cosmology because it provides a means of testing models which assume different things about the contents of the universe See also editRipley s K and Besag s L function Correlation function in statistics Spatial point processReferences editPeebles P J E 1980 The large scale structure of the universe Theuns Physical Cosmology Retrieved from https en wikipedia org w index php title Correlation function astronomy amp oldid 1176436556, wikipedia, wiki, book, books, library,
