Three-point Correlation Function Of Galaxies Research Articles

The Analytical Equation of the Three-point Correlation Function of Galaxies: to the Third Order of Density Perturbation

Applying functional differentiation to the density field with Newtonian gravity, we obtain the static, nonlinear equation of the three-point correlation function ζ of galaxies to the third order density perturbations. We make the equation closed and perform renormalization of the mass and the Jeans wavenumber. Using the boundary condition inferred from observations, we obtain the third order solution ζ(r, u, θ) at fixed u = 2, which is positive, exhibits a U-shape along the angle θ, and decreases monotonously along the radial r up to the range r ≤ 30 h −1 Mpc in our computation. The corresponding reduced Q(r, u, θ) deviates from 1 of the Gaussian case, has a deeper U-shape along θ, and varies non-monotonously along r. The third order solution agrees with the SDSS data of galaxies, quite close to the previous second order solution, especially at large scales. This indicates that the equations of correlation functions with increasing orders of density perturbation provide a stable description of the nonlinear galaxy system.

The Nonlinear Field Equation of the Three-point Correlation Function of Galaxies: to the Second Order of Density Perturbation

Based on the field theory of density fluctuation under Newtonian gravity, we obtain analytically the nonlinear equation of 3-pt correlation function ζ of galaxies in a homogeneous, isotropic, static universe. The density fluctuation has been kept up to second order. By the Fry–Peebles ansatz and the Groth-Peebles ansatz, the equation of ζ becomes closed and differs from the Gaussian approximate equation. Using the boundary condition inferred from the data of SDSS, we obtain the solution ζ(r, u, θ) at fixed u = 2, which exhibits a shallow U-shape along the angle θ and, nevertheless, decreases monotonously along the radial r. We show its difference with the Gaussian solution. As a direct criterion of non-Gaussianity, the reduced Q(r, u, θ) deviates from the Gaussianity plane Q = 1, exhibits a deeper U-shape along θ and varies weakly along r, agreeing with the observed data.

Exploring the squeezed three-point galaxy correlation function with generalized halo occupation distribution models

We present the GeneRalized ANd Differentiable Halo Occupation Distribution (GRAND-HOD) routine that generalizes the standard 5 parameter halo occupation distribution model (HOD) with various halo-scale physics and assembly bias. We describe the methodology of 4 different generalizations: satellite distribution generalization, velocity bias, closest approach distance generalization, and assembly bias. We showcase the signatures of these generalizations in the 2-point correlation function (2PCF) and the squeezed 3-point correlation function (squeezed 3PCF). We identify generalized HOD prescriptions that are nearly degenerate in the projected 2PCF and demonstrate that these degeneracies are broken in the redshift-space anisotropic 2PCF and the squeezed 3PCF. We also discuss the possibility of identifying degeneracies in the anisotropic 2PCF and further demonstrate the extra constraining power of the squeezed 3PCF on galaxy-halo connection models. We find that within our current HOD framework, the anisotropic 2PCF can predict the squeezed 3PCF better than its statistical error. This implies that a discordant squeezed 3PCF measurement could falsify the particular HOD model space. Alternatively, it is possible that further generalizations of the HOD model would open opportunities for the squeezed 3PCF to provide novel parameter measurements. The GRAND-HOD Python package is publicly available at https://github.com/SandyYuan/GRAND-HOD .

Computing the three-point correlation function of galaxies in $\mathcal {O}(N^2)$ time

We present an algorithm that computes the multipole coefficients of the galaxy three-point correlation function (3PCF) without explicitly considering triplets of galaxies. Rather, centering on each galaxy in the survey, it expands the radially-binned density field in spherical harmonics and combines these to form the multipoles without ever requiring the relative angle between a pair about the central. This approach scales with number and number density in the same way as the two-point correlation function, allowing runtimes that are comparable, and 500 times faster than a naive triplet count. It is exact in angle and easily handles edge correction. We demonstrate the algorithm on the LasDamas SDSS-DR7 mock catalogs, computing an edge corrected 3PCF out to $90\;{\rm Mpc}/h$ in under an hour on modest computing resources. We expect this algorithm will render it possible to obtain the large-scale 3PCF for upcoming surveys such as Euclid, LSST, and DESI.

Accelerating the two-point and three-point galaxy correlation functions using Fourier transforms

Abstract Though Fourier transforms (FTs) are a common technique for finding correlation functions, they are not typically used in computations of the anisotropy of the two-point correlation function (2PCF) about the line of sight in wide-angle surveys because the line-of-sight direction is not constant on the Cartesian grid. Here we show how FTs can be used to compute the multipole moments of the anisotropic 2PCF. We also show how FTs can be used to accelerate the 3PCF algorithm of Slepian & Eisenstein. In both cases, these FT methods allow one to avoid the computational cost of pair counting, which scales as the square of the number density of objects in the survey. With the upcoming large data sets of Dark Energy Spectroscopic Instrument, Euclid, and Large Synoptic Survey Telescope, FT techniques will therefore offer an important complement to simple pair or triplet counts.

On the signature of the baryon–dark matter relative velocity in the two- and three-point galaxy correlation functions

We develop a configuration-space picture of the relative velocity between baryons and dark matter that clearly explains how it can shift the BAO scale in the galaxy-galaxy correlation function. The shift occurs because the relative velocity is non-zero only within the sound horizon and thus adds to the correlation function asymmetrically about the BAO peak. We further show that in configuration space the relative velocity has a localized, distinctive signature in the three-point galaxy correlation function (3PCF). In particular, we find that a multipole decomposition is a favorable way to isolate the relative velocity in the 3PCF, and that there is a strong signature in the l=1 multipole for triangles with 2 sides around the BAO scale. Finally, we investigate a further compression of the 3PCF to a function of only one triangle side that preserves the localized nature of the relative velocity signature while also nicely separating linear from non-linear bias. We expect that this scheme will substantially lessen the computational burden of finding the relative velocity in the 3PCF. The relative velocity's 3PCF signature can be used to correct the shift induced in the galaxy-galaxy correlation function so that no systematic error due to this effect is introduced into the BAO as used for precision cosmology.

Statistical analysis of galaxy surveys — II. The three-point galaxy correlation function measured from the 2dFGRS

We present new results for the 3-point correlation function, �, measured as a function of scale, luminosity and colour from the final version of the two-degree field galaxy redshift survey (2dFGRS). The reduced three point correlation function, Q3 � �/� 2 , is estimated for different triangle shapes and sizes, employing a full covariance analysis. The form of Q3 is consistent with the expectations for the �-cold dark matter model, confirming that the primary influence shaping the distribution of galaxies is gravitational instability acting on Gaussian primordial fluctuations. However, we find a clear offset in amplitude between Q3 for galaxies and the predictions for the dark matter. We are able to rule out the scenario in which galaxies are unbiased tracers of the mass at the 9-� level. On weakly non-linear scales, we can interpret our results in terms of galaxy bias parameters. We find a linear bias term that is consistent with unity, b1 = 0.93 +0.10 0.08 and a quadratic bias c2 = b2/b1 = 0.34 +0.11 0.08 . This is the first significant detection of a non-zero quadratic bias, indicating a small but important non-gravitational contribution to the three point function. Our estimate of the linear bias from the three point function is independent of the normalisation of underlying density fluctuations, so we can combine this with the measurement of the power spectrum of 2dFGRS galaxies to constrain the amplitude of matter fluctuations. We find that the rms linear theory variance in spheres of radius 8h 1 Mpc is �8 = 0.88 +0.12 0.10, providing an independent confirmation of values derived from other techniques. On non-linear scales, where � > 1, we find that Q3 has a strong dependence on scale, colour and luminosity.

The three-point correlation function of galaxies: comparing halo occupation models with observations

We present models for the three-point correlation function (3PCF) of both dark matter and galaxies. We show that models based on the halo model can reasonably match the dark-matter 3PCF obtained from high-resolution N-body simulations. On small scales (r ≤ 0.5 h -1 Mpc) the 3PCF is sensitive to details regarding the density distributions of dark-matter haloes. On larger scales (r ≥ 2.0 h -1 Mpc) the results are very sensitive to the abundance of the few most prominent haloes. Using the conditional luminosity function, we also construct models for the 3PCF of galaxies, which we test against large mock galaxy samples. The bias of the galaxy distribution with respect to the dark matter, and the finite number of galaxies that can be hosted by individual haloes, significantly reduce the normalized three-point correlation function with respect to that of dark matter. Contrary to the 3PCF of the dark matter, the galaxy 3PCF is much less sensitive to details regarding the spatial number density distribution of galaxies in individual haloes or to the abundance of the few most massive systems. Finally, we show that our model based on the conditional luminosity function is in good agreement with results obtained from the 2-degree Field Galaxy Redshift Survey. In particular, the model nicely reproduces the observational finding that the 3PCF for early-type galaxies is slightly higher than that of late-type galaxies, and that there is no significant dependence of the 3PCF on galaxy luminosity.

Three-Point Correlation Functions of SDSS Galaxies in Redshift Space: Morphology, Color, and Luminosity Dependence

Abstract We present measurements of the redshift-space three-point correlation function of galaxies in the Sloan Digital Sky Survey (SDSS). For the first time, we analyze the dependence of this statistic on galaxy morphology, color, and luminosity. In order to control systematics due to selection effects, we used $r$-band, volume-limited samples of galaxies, constructed from magnitude-limited SDSS data ($14.5 < r < 17.5$), and further divided the samples into two morphological types (early and late) or two color populations (red and blue). The three-point correlation function of SDSS galaxies follows the hierarchical relation well, and the reduced three-point amplitudes in redshift-space are almost scale-independent ($Q_z = 0.5 \sim 1.0$). In addition, their dependence on the morphology, color, and luminosity is not statistically significant. Given the robust morphological, color, and luminosity dependences of the two-point correlation function, this implies that galaxy biasing is complex on weakly non-linear to non-linear scales. We show that a simple deterministic linear relation with the underlying mass could not explain our measurements on these scales.

The Projected Three-Point Correlation Function: Theory and Observations

We report results for the angular three-point galaxy correlation function in the APM survey and compare them with theoretical expectations. For the first time, these measurements extend to sufficiently large scales to probe the weakly non-linear regime. On large scales, the results are in good agreement with the predictions of non-linear cosmological perturbation theory, for a model with initially Gaussian fluctuations and linear power spectrum $P(k)$ consistent with that inferred from the APM survey. These results reinforce the conclusion that large-scale structure is driven by non-linear gravitational instability and that APM galaxies are relatively unbiased tracers of the mass on large scales; they also provide stringent constraints upon models with non-Gaussian initial conditions and strongly exclude the standard cold dark matter model.

The second and third moments of Abell clusters and their three-point correlation function

Using the method of counts in cells we calculated the second and third moments in the angular distribution of northern Abell clusters with richness classes ⩾ 1 and hence their spatial three-point correlation. We also calculated the three-point correlation function for a redshift sample of D⩽4, R⩾1 Abell clusters, the results are in agreement with those obtained for samples with estimated redshifts, thus showing further that the three-point correlation function of galaxies and Abell clusters can be described by a universal form.

Theoretical forms of two- and three-point galaxy correlation functions

The power law form of the two-point galaxy correlation function ξ, and a relationship between ξ and the three-point correlation function ζ, are derived theoretically. These derivations provide a theoretical basis for expressions that heretofore have been based solely upon observations of galaxy or cluster clustering.

Observational constraints on dark matter in the universe

Application of the cosmic virial theorem to galaxy redshift surveys suggests an open universe - if the dark material is distributed in the same way as galaxies. Here we investigate four redshift surveys. We present new results on the amplitudes of the two- and three-point galaxy correlation functions and we estimate the relative peculiar velocities in the deep redshift survey of Kirshner and co-workers (1983). Next, we review evidence which suggests that dark material is clustered on scales comparable to the optical radii of galaxies. In particular, we use spectroscopic and photometric observations to set constraints on the mass distribution in elliptical galaxies.

Transform analysis of the high-resolution Shane-Wirtanen Catalog - The power spectrum and the bispectrum

To extract more statistical information from the high resolution (10') Shane-Wirtanen data, and in particular to gain information about the large-scale behavior of galaxy clustering, we have examined the two- and three-point galaxy correlation functions in Fourier transform space for wavelengths corresponding to 1/2 /sup 0/< or approx. =theta< or approx. =70/sup 0/ (2 Mpc< or approx. =hr< or approx. =260 Mpc, h = Hubble's constant in units of 100 km s/sup -1/ Mpc/sup -1/). To the extent that our results can be compared to earlier works, we obtain rough agreement, but we are able to identify some systematic differences also. There is no evidence for any transition in behavior of the power spectrum corresponding to the break in the two-point function at 2./sup 0/5 (9 h/sup -1/ Mpc), but this characteristic scale does appear in the bispectrum. The small k (large angle) behavior of the power spectrum is as k/sup n/, with nroughly-equal-1.

A cosmic virial theorem

A new type of cosmic virial theorem is derived, representing a relation among the two-point and three-point galaxy correlation functions and the mean square relative peculiar velocity 〈v 2(r)〉 averaged over pairs of galaxies separated by distancer. The two correlation functions seem to be fairly well known, but 〈v 2(r)〉 at best is only very roughly estimated from available data. It might be possible to improve the estimate through a systematic program of redshift measurements in sample areas. If so, the cosmic virial theorem might yield an interesting measure of the mean mass density due to galaxies.

A Cosmic Virial Theorem

A new type of cosmic virial theorem is derived, representing a relation among the two-point and three-point galaxy correlation functions and the mean square relative peculiar velocity 〈v 2(r)〉 averaged over pairs of galaxies separated by distancer. The two correlation functions seem to be fairly well known, but 〈v 2(r)〉 at best is only very roughly estimated from available data. It might be possible to improve the estimate through a systematic program of redshift measurements in sample areas. If so, the cosmic virial theorem might yield an interesting measure of the mean mass density due to galaxies.