A statistical tool that measures the clustering of matter by comparing the probability of finding pairs at given separations.The correlation function is a statistical measure used in cosmology and astrophysics to quantify how matter is clustered in the universe. It compares the probability of finding two objects (like galaxies) at a given separation against what you would expect if matter were distributed randomly. A positive value means objects are more likely to be found near each other than chance would suggest, indicating clustering; a negative value means they are less likely, indicating avoidance.
The two-point correlation function was introduced into cosmology in the 1970s, notably by James Peebles and his collaborators, who used it to analyze galaxy catalogs.
To compute it, you take a catalog of objects and count the number of pairs at each separation distance. You then divide by the number of pairs expected from a random distribution with the same density. The result is often expressed as a function of separation, showing how clustering strength changes with scale. On large scales, the correlation function follows a power law, reflecting the hierarchical structure of the universe.
It is used to measure the distribution of galaxies and dark matter, helping constrain cosmological parameters like the matter density and the nature of dark energy.
The correlation function of the cosmic microwave background radiation was a key piece of evidence for the existence of dark matter.
| K | variable | Leggett-Garg combination of correlation functions |
| C_{ab} | variable | Correlation function between measurements at times a and b — correlation function |
| C_{a'b} | variable | Correlation function between measurements at times a' and b — correlation function |
| C_{ab'} | variable | Correlation function between measurements at times a and b' — correlation function |
| C_{a'b'} | variable | Correlation function between measurements at times a' and b' — correlation function |
| b | variable | bias parameter — bias parameter |
| ξ_gal | variable | galaxy two-point correlation function — correlation function |
| ξ_DM | variable | dark matter two-point correlation function — correlation function |
| √ | operator | takes the square root of the ratio — square root |
| C(t) | variable | Normalized autocorrelation function of observable A — correlation function |
| A(t) | variable | Value of observable A at time t — observable varies |
| A(0) | variable | Value of observable A at initial time — observable varies |
| ⟨...⟩ | operator | Averages the product over a statistical ensemble of initial conditions — ensemble average |
| C(\omega,\omega') | variable | Correlation function of fluctuations at two frequencies — correlation function |
| H(k;\omega) | variable | Transfer function or response kernel at wavevector k and frequency omega — transfer function |
| H^*(k;\omega') | variable | Complex conjugate of the transfer function at frequency omega' — transfer function |
| P(k) | variable | Power spectrum of the underlying fluctuations — power spectrum |
| \int dk | operator | Integration over all wavevectors — integral operator |
| F | variable | Correlation function — correlation function |
| F_0 | variable | Normalization constant — correlation function |
| C_QE | variable | Degree of quantum entanglement — entanglement degree |
| A(\theta_1) | variable | Angular modulation factor at angle theta_1 — modulation factor |
| A(\theta_2) | variable | Angular modulation factor at angle theta_2 — modulation factor |
| \theta_1 | variable | Scattering angle for first particle — angle radian |
| \theta_2 | variable | Scattering angle for second particle — angle radian |
| \phi | variable | Azimuthal angle — angle radian |
| \cos | operator | Trigonometric cosine function — cosine |
