n-Point Correlation Functions: Descriptive Statistics

2020 ◽  
pp. 138-256
Author(s):  
P. J. E. Peebles
Keyword(s):  
Correlation Functions ◽  
Practical Importance ◽  
Descriptive Statistics ◽  
Statistical Estimates ◽  
Galaxy Distribution ◽  
Statistical Pattern ◽  
The Galaxy ◽  
Angular Correlation Function ◽  
Point Correlation ◽  
Simple Linear Equation

This chapter explores the statistical pattern of the galaxy distribution. It focuses on n-point correlation functions (analogs of the autocorrelation function and higher moments for a continuous function), the descriptive statistics that have proved useful. The approach has also proved useful in many other applications. Of considerable practical importance has been the fact that there is a simple linear equation relating the directly observable angular correlation function to the wanted spatial function. This means the translation from one to the other is fairly easy, and equally important it makes it easy to say how the statistical estimates ought to scale with the depth of the survey and hence to test for possible contamination of the estimates by systematic errors. A third useful result is that the dynamics of the galaxy distribution can be treated in terms of the mass correlation functions: the statistic that proves useful for the reduction of the data may also be useful for the analysis of the theory.

scholarly journals N-body Simulations to Test the Reliability of Two-point Correlation Functions of Galaxies

1992 ◽  
Vol 9 ◽  
pp. 703-704
Author(s):  
Yasushi Suto
Keyword(s):  
Correlation Functions ◽  
Large Scale ◽  
Sample Volume ◽  
Limited Data ◽  
Observable Universe ◽  
Large N ◽  
The Galaxy ◽  
Point Correlation ◽  
Systematic Biases ◽  
Redshift Survey

The shape and amplitude of the galaxy – galaxy correlation functions, ξgg(r), are among the most widely used measures of the large-scale structure in the universe (Totsuji & Kihara 1969). The estimates, however, might be seriously affected by the limited size of the sample volume, or equivalently, the limited number of available galaxies. In fact, while the observable universe extends c/H0 ~ 3000h-1Mpc, most observational works to map the distribution of galaxies so far have been mainly applied to samples within ~ 100h-1Mpc from us. Thus a CfA redshift survey slice, for example, of 8h < α < 17h, 26.5° < δ < 32.5°, and cz ≾ 15000km/sec (de Lapparent et al. 1986, 1988) represents merely ~ 2 x 10-5 of the total volume of the observable universe. This clearly illustrates the importance of examining possible systematic biases and variations in the estimates of two-point correlation functions from instrinsically limited data. We studied such sample-to-sample variations by analysing subsamples extracted from large N-body simulation data.

scholarly journals The Void Probability Function as a Statistical Indicator

1987 ◽  
Vol 124 ◽  
pp. 359-362
Author(s):  
Marc Lachièze-Rey ◽  
Sophie Maurogordato
Keyword(s):  
Correlation Functions ◽  
Statistical Models ◽  
Probability Function ◽  
Statistical Indicator ◽  
Galaxy Distribution ◽  
The Galaxy ◽  
Statistical Indicators

Recent results concerning the galaxy distribution at scales < 100 h−1 Mpc (Ho = 100 h kms−1 Mpc−1) show a number of characteristics which cannot be described by conventional statistical models. Correlation functions, for instance, can in no way give account of the presence of voids of the cellular (or spongy) appearance of the local galaxy distribution (M. Geller, this conference). There is clearly a need for new kinds of statistical models and statistical indicators.

scholarly journals The autocorrelation analysis of deep galaxy samples

1978 ◽  
Vol 79 ◽  
pp. 280-280
Author(s):  
S. Phillipps
Keyword(s):  
Correlation Function ◽  
Small Area ◽  
Angular Diameter ◽  
Selection Function ◽  
Autocorrelation Analysis ◽  
Galaxy Distribution ◽  
Royal Observatory ◽  
The Galaxy ◽  
Point Correlation ◽  
Point Correlation Function

The two point correlation function w(θ) has been evaluated for the galaxies measured by the COSMOS machine at the Royal Observatory, Edinburgh, in an area of about 2 square degrees on a 2 hour exposure J plate and a 2 hour exposure R plate (Phillipps, S., Fong, R., Ellis, R.S., Fall, S.M. and MacGillivray, H.T., 1977, Mon. Not. R. astr. Soc., in press). in each case w(θ) is found to be in agreement with the form w = Aθ−0.8 found previously by Peebles and coworkers. Since the samples are not magnitude limited the selection function, i.e. the distribution in distance, was determined by using models of the galaxy distribution to fit the observed angular diameter counts. However, when these selection functions are used to scale the amplitudes found for our samples, the amplitudes are found to be lower than those expected from Peebles' results by a factor of about 3. We consider that this is likely to be due to a lack of clusters in the small area of sky which we have studied: analysis of further areas should show whether this is the case.

Dynamics and Statistics

2020 ◽  
pp. 257-303
Author(s):  
P. J. E. Peebles
Keyword(s):  
Correlation Functions ◽  
Bbgky Hierarchy ◽  
Dynamical Theory ◽  
Descriptive Statistics ◽  
Expanding Universe ◽  
Newtonian Theory ◽  
Theory Of Evolution ◽  
Joint Distributions ◽  
Point Correlation ◽  
Dynamic Variables

This chapter studies how the n-point correlation functions have proved useful not only as descriptive statistics but also as dynamic variables in the Newtonian theory of the evolution of clustering. It generalizes the functions to mass correlation functions in position and momentum, and derives the BBGKY hierarchy of equations for their evolution. This yields a new way to analyze the evolution of mass clustering in an expanding universe. Of course, the main interest in the approach comes from the thought that the observed galaxy correlation functions may yield useful approximations to the mass correlation functions, so the observations may provide boundary values for the dynamical theory of evolution of the mass correlation functions. The test will be whether one can find a consistent theory for the joint distributions in galaxy positions and velocities.

scholarly journals On the Constraints of Galaxy Assembly Bias in Velocity Space

2021 ◽  
Author(s):  
Kevin S McCarthy ◽  
Zheng Zheng ◽  
Hong Guo ◽  
Wentao Luo ◽  
Yen-Ting Lin
Keyword(s):  
Dark Matter ◽  
Correlation Functions ◽  
Real Space ◽  
Velocity Space ◽  
Bias Effect ◽  
The Galaxy ◽  
History Of ◽  
Point Correlation ◽  
Galaxy Assembly ◽  
Velocity Bias

Abstract If the formation of central galaxies in dark matter haloes traces the assembly history of their host haloes, in haloes of fixed mass, central galaxy clustering may show dependence on properties indicating their formation history. Such a galaxy assembly bias effect has been investigated by Lin et al. (2016), with samples of central galaxies constructed in haloes of similar mass and with mean halo mass verified by galaxy lensing measurements, and no significant evidence of assembly bias is found from the analysis of the projected two-point correlation functions of early- and late-forming central galaxies. In this work, we extend the the investigation of assembly bias effect from real space to redshift (velocity) space, with an extended construction of early- and late-forming galaxies. We carry out halo occupation distribution modelling to constrain the galaxy-halo connection to see whether there is any sign of the effect of assembly bias. We find largely consistent host halo mass for early- and late-forming central galaxies, corroborated by lensing measurements. The central velocity bias parameters, which are supposed to characterise the mutual relaxation between central galaxies and their host haloes, are inferred to overlap between early- and late-forming central galaxies. However, we find a large amplitude of velocity bias for early-forming central galaxies (e.g. with central galaxies moving at more than 50% that of dark matter velocity dispersion inside host haloes), which may signal an assembly bias effect. A large sample with two-point correlation functions and other clustering measurements and improved modelling will help reach a conclusive result.

scholarly journals Evidence for the gravitational instability picture in a dense Universe

1978 ◽  
Vol 79 ◽  
pp. 441-443
Author(s):  
Marc Davis
Keyword(s):  
Correlation Function ◽  
Random Distribution ◽  
Initial Conditions ◽  
Gravitational Instability ◽  
Statistical Nature ◽  
Galaxy Distribution ◽  
The Galaxy ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Recombination Epoch

The statistical nature of the galaxy distribution is in a sense remarkably simple. the two-point correlation function ξ(r), which measures the count of galaxies at separation r in excess of that expected for a random distribution, varies as ξ∝;r−1.8 for ξ>1(r ≲15 Mpc). At larger separations ξ apparently decreases more rapidly. the power law behavior is observed in different galaxy catalogs of varying depth and positions in the sky. What is the explanation of this universal behavior of ξ(r), and what do correlation functions tell us about the initial conditions at the recombination epoch and/or the value of Ω?

scholarly journals THE FREE FIELD REPRESENTATION OF SU(3) CONFORMAL FIELD THEORY

1993 ◽  
Vol 08 (23) ◽  
pp. 4031-4053
Author(s):  
HOVIK D. TOOMASSIAN
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Correlation Functions ◽  
Free Field ◽  
Field Representation ◽  
Conformal Field ◽  
Point Correlation

The structure of the free field representation and some four-point correlation functions of the SU(3) conformal field theory are considered.

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