这是一篇对2pcf及功率谱的简要介绍。由于学术书面语言需要，下文使用英语。/

Here is a short introduction of 2pcf and power spectrum. The following would be written in English.

Correlation function

2PCF

  • : the number of pairs in volumes and seperated by .

    Def 2pcf (2 point correlation function , is the average numerical density.

​ Obviously, .

  • , average, may have 2 meanings.

    • taking many relizations of the distribution, all of them produced in the same IC (initial condition), selecting in each realization that at the same locations an then aceraging the pair number — ensemble average.
    • take pairs at different spots within the same realization seperated by the same .

  • If we take the average as the sample average, then .

  • In practice, we often choose s.t. . Then we have the follows:

  • i.e. evaluates the correlation function as the average number of particles at distant from any given particle, divided by the expected number of particles at the same distance in a uniform distribution, minus 1.

    In a finite volume with particles, apparently.

  • If positive, there are more particles than in a uniform distribution, i.e. the distribution is positively clustered.

  • Usually only consider its dependence on the modulus . So the volume is chosen as a shell around each particle. Then is the number density inside a shell of thickness at distance from every particle. Just requires the estimation of the density in every shell.

    • One could generalize the Def by introducing anisotropic CF as # of pairs in V separated by vec. .

  • The simplest way to measure is to compare the real catalog to an artificial random catalog with exactly the same boundaries and the same selection function.

    is the number density of galaxies in the data and random catalogs. means the number of galaxies at distance r counted by an observer centered on a real galaxy (data ). is the number of galaxies at the same distance but in the random catalog centered on a real galaxy. Actually we are estimating by counting galaxies with MC method.

  • “random catalog” has the identical three dimensional coverage as the data - including the same sky coverage and smoothed redshift distribution - but is populated with randomly-distribution points.

  • An estimator with smaller err by Hamilton 1993:

  • A most commonly-used estimator is from Landy & Szalay (1993) with smaller errors:

    It’s less sensitive to the size of random catalog and handles edge corrections well.

nPCF

  • 1pcf — average number density ; 2pcf –

  • 3pcf:

    Then

    When , a field is Gaussian, then the 2pcf or the power spectrum completely describes the statistical properties of the field.

Power spectrum

Power spectrum is the FT of 2pcf:

Wavelengths in different modes are uncorrelated if the field is statistically homogeneous.

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