Correlation function (astronomy)
Encyclopedia
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 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.
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, a correlation function describes the distribution of galaxies
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...
in the universe. By default, correlation function refers to the two-point autocorrelation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
. For a given distance, the two-point autocorrelation function is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of one variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
(distance) which describes the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
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 probabilityProbabilityProbability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" 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 function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
.
The n-point autocorrelation functions for n greater than 2 or cross-correlation
Cross-correlation
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...
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
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
because it provides a means of testing models which assume different things about the contents of the universe.