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

With the advent of high-quality surveys in cosmology, the full three-point correlation function (3PCF) will be a valuable statistic for describing structure formation models. It contains information on cosmological parameters and detailed halo properties that cannot be extracted from the two-point correlation function. We use the halo clustering model to calculate the three-point correlation function analytically for general cosmological fields. We present detailed results for the configuration dependence of the three-dimensional mass and galaxy distributions and the two-dimensional cosmic shear field. We work in real space, where higher order correlation functions on small scales are easier to measure and interpret, but halo model calculations become rapidly intractable. Hence we develop techniques that allow for the accurate evaluation of all the contributing terms to real-space correlations. We apply them to the 3PCF to show how its configuration and scale dependence change as one transits from the non-linear to the quasilinear regime, and how this depends on the relative contributions from the one-, two- and three-halo terms. The 3PCF violates the hierarchical Ansatz in both its scale and configuration dependence. We study the behaviour of the coefficient Q in the expansion: ζ(r 1 2 , r 2 3 , r 3 1 ) = Q[ξ(r 1 2 ) ξ(r 2 3 ) + ξ(r 1 2 )ξ(r 3 1 ) + ξ(r 2 3 )ξ(r 3 1 )] from large, quasilinear scales down to approximately 20 kpc. We find that the non-linear 3PCF is sensitive to the halo profile of massive haloes, especially its inner slope. We model the distribution of galaxies in haloes and show that the 3PCF of red galaxies has a weaker configuration and scale dependence than the mass, while for blue galaxies it is very sensitive to the parameters of the galaxy formation model. The 3PCF from weak lensing on the other hand shows different scalings arising from projection effects and a sensitivity to cosmological parameters. We discuss how our results can be applied to various analytical calculations: covariances of the two-point correlation function, the pairwise peculiar velocity dispersion, higher order shear correlations, and to extend the halo model by including the effects of halo triaxiality and substructure on statistical measures.

We present measurements of the redshift-space three-point correlation function of 50 967 luminous red galaxies (LRGs) from Data Release 3 (DR3) of the Sloan Digital Sky Survey (SDSS). We have studied the shape dependence of the reduced three-point correlation function (Qz(s, q, θ)) on three different scales, s= 4, 7 and 10 h−1 Mpc, and over the range of 1 < q < 3 and 0° < θ < 180°. On small scales (s= 4 h−1 Mpc), Qz is nearly constant, with little change as a function of q and θ. However, there is evidence for a shallow U-shaped behaviour (with θ) which is expected from theoretical modelling of Qz(s, q, θ). On larger scales (s= 7 and 10 h−1 Mpc), the U-shaped anisotropy in Qz (with θ) is more clearly detected. We compare this shape dependence in Qz(s, q, θ) with that seen in mock galaxy catalogues which were generated by populating the dark matter haloes in large N-body simulations with mock galaxies using various halo occupation distributions (HOD). We find that the combination of the observed number density of LRGs, the (redshift-space) two-point correlation function and Qz(s, q, θ) provides a strong constraint on the allowed HOD parameters (Mmin, M1, α) and breaks key degeneracies between these parameters. For example, our observed Qz(s, q, θ) disfavours mock catalogues that overpopulate massive dark matter haloes with many LRG satellites. We also estimate the linear bias of LRGs to be b= 1.87 ± 0.07 in excellent agreement with other measurements.

The three-point correlation function (3PCF) provides an important view into the clustering of galaxies that is not available to its lower order cousin, the two-point correlation function (2PCF). Higher order statistics, such as the 3PCF, are necessary to probe the non-Gaussian structure and shape information expected in these distributions. We measure the clustering of spectroscopic galaxies in the Main Galaxy Sample of the Sloan Digital Sky Survey (SDSS), focusing on the shape or configuration dependence of the reduced 3PCF in both redshift and projected space. This work constitutes the largest number of galaxies ever used to investigate the reduced 3PCF, using over 220,000 galaxies in three volume-limited samples. We find significant configuration dependence of the reduced 3PCF at 3-27 Mpc/h, in agreement with LCDM predictions and in disagreement with the hierarchical ansatz. Below 6 Mpc/h, the redshift space reduced 3PCF shows a smaller amplitude and weak configuration dependence in comparison with projected measurements suggesting that redshift distortions, and not galaxy bias, can make the reduced 3PCF appear consistent with the hierarchical ansatz. The reduced 3PCF shows a weaker dependence on luminosity than the 2PCF, with no significant dependence on scales above 9 Mpc/h. On scales less than 9 Mpc/h, the reduced 3PCF appears more affected by galaxy color than luminosty. We demonstrate the extreme sensitivity of the 3PCF to systematic effects such as sky completeness and binning scheme, along with the difficulty of resolving the errors. Some comparable analyses make assumptions that do not consistently account for these effects.

Aims. We describe various expansion schemes that can be used to study gravitational clustering. Obtained from the equations of motion or their path-integral formulation, they provide several perturbative expansions that are organized in a different fashion or involve different partial resummations. We focus on the two-point and three-point correlation functions, but these methods also apply to all higher-order correlation and response functions.Methods. We present the general formalism, which holds for the gravitational dynamics and for any similar models, such as the Zeldovich dynamics, that obey similar hydrodynamical equations of motion with a quadratic nonlinearity. We give our explicit analytical results up to one-loop order for the simpler Zeldovich dynamics. For the gravitational dynamics, we compare our one-loop numerical results with numerical simulations.Results. We check that the standard perturbation theory is recovered from the path integral by expanding over Feynman's diagrams. However, the latter expansion is organized in a different fashion, and it contains some UV divergences that cancel out as we sum all diagrams of a given order. Resummation schemes modify the scaling of tree and one-loop diagrams, which exhibit the same scaling over the linear power spectrum (contrary to the standard expansion). However, they do not significantly improve over standard perturbation theory for the bispectrum, unless one uses accurate two-point functions (e.g. a fit to the nonlinear power spectrum from simulations). Extending the range of validity to smaller scales, to reach the range described by phenomenological models, seems to require at least two-loop diagrams.

We study anisotropic (patchy) screening induced by the resonant conversion of cosmic microwave background (CMB) photons into dark-sector massive vector bosons (dark photons) as they cross non-linear large scale structure (LSS). Resonant conversion takes place through the kinetic mixing of the photon with the dark photon, one of the simplest low energy extensions to the Standard Model. In the early Universe, resonant conversion can occur when the photon plasma mass, obtained as the photon propagates through the ionized interstellar and intergalactic media, matches the dark photon mass. After the epoch of reionization, resonant conversion occurs mainly in the ionized gas that occupies virialized dark matter halos, for a range of dark photon masses between 10-13 eV ≲ m A' ≲ 10-11 eV. This leads to new CMB anisotropies that are correlated with LSS, which we refer to as patchy dark screening, in analogy with anisotropies from Thomson screening. Its unique frequency dependence allows it to be distinguished from the blackbody CMB. In this paper, we use a halo model approach to predict the imprint of dark screening on the CMB temperature and polarization anisotropies, as well as their correlation with LSS. We then examine the two- and three-point correlation functions of the dark-screened CMB, as well as correlation functions between CMB and LSS observables, to project the sensitivity of future measurements to the kinetic mixing parameter and dark photon mass. We demonstrate that an analysis with existing CMB data can improve upon current constraints on the kinetic mixing parameter by two orders of magnitude with the two-point correlation functions, while data from upcoming CMB experiments and LSS surveys can further improve the reach by another order of magnitude with two- and three-point correlation functions.

view Abstract Citations (28) References (44) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Cluster-Cluster Correlation in Hybrid Models Jing, Y. P. ; Mo, H. J. ; Boerner, G. ; Fang, L. Z. Abstract Using N-body simulations, we have calculated the two-point correlation functions of the clusters of galaxies (ξ_cc_) in a set of hybrid (CDM + HDM + baryon) models. The galaxies are identified as density peaks above certain thresholds, to match the observed two-point correlation function of galaxies on scales <~10 h^-1^ Mpc. The clusters of galaxies are identified such that their number density is the same as that of Abell clusters. It is found that both the coherence length (the largest scale where ξ_cc_ remains positive) and the amplitude of ξ_cc_ increase systematically with the power of the initial density spectrum on large scales. For one model in which the power on large scales is sufficient to match the two-point angular correlation function of galaxies of the APM survey, ξ_cc_has a correlation length r_0_ = 23.0 +/- 0.9 h^-1^ Mpc and remains positive for r <~70 h^-1^ Mpc with a weak dependence on the richness of clusters, in good agreement with observations. Our results suggest that the observed large correlation length of Abell clusters, which is sometimes considered to be due to contaminations, should be expected in models which have enough power to explain the recent observations of galaxy clustering and motions on large scales. Publication: The Astrophysical Journal Pub Date: July 1993 DOI: 10.1086/172847 Bibcode: 1993ApJ...411..450J Keywords: Angular Correlation; Astronomical Models; Dark Matter; Galactic Clusters; Interacting Galaxies; Many Body Problem; Galactic Evolution; Galactic Structure; Universe; Astrophysics; COSMOLOGY: DARK MATTER; GALAXIES: CLUSTERING; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; METHODS: NUMERICAL full text sources ADS |

The three-point correlation function (3PCF) of the 21cm brightness temperature from the Epoch of Reionization (EoR) probes complementary information to the commonly studied two-point correlation function (2PCF) about the morphology of ionized regions. We investigate the 21cm 2PCF and 3PCF in configuration space using semi-numerical simulations and test whether they can be described by the local quadratic bias model. We find that fits of bias model predictions for the 2PCF and 3PCF deviate from our measurements by $\sim 20\%$ at scales above the typical size of ionized regions ($\simeq 30$ Mpc) and at early times with global neutral fractions of $\langle x_{\rm HI} \rangle \gtrsim 0.7$. At later times and smaller scales these deviations increase strongly, indicating a break down of the bias model. The 2PCF and 3PCF fits of the linear bias parameter agree at the $10\%$ level for different EoR model configurations. This agreement holds, when adding redshift space distortions to the simulations. The relation between spatial fluctuations in the matter density and the 21cm signal, as predicted by the bias model, is consistent with direct measurements of this relation in simulations for large smoothing scales ($\gtrsim 30$ Mpc). From this latter test we conclude that negative amplitudes of the 21cm 3PCF result from negative bias parameters, which describe the anti-correlation between the matter over-densities and the 21cm signal during the EoR. However, a more detailed interpretation of the bias parameters may require a description of non-local contributions to the bias model.

We perform a model-independent, non-perturbative investigation of two-point and three-point finite-volume correlation functions in the energy regime where two-particle states can go on-shell. We study three-point functions involving a single incoming particle and an outgoing two-particle state, relevant, for example, for studies of meson decays (e.g., B-to-pi Kll) or meson photo production (e.g., pi gamma-to-pi pi). We observe that, while the spectrum solely depends on the on-shell scattering amplitude, the correlation functions also depend on off-shell amplitudes. The main result of this work is a generalization of the Lellouch-Luscher formula relating matrix elements of currents in finite and infinite spatial volumes. We extend that work by considering a theory with multiple, strongly-coupled channels and by accommodating external currents which inject arbitrary four-momentum as well as arbitrary angular momentum. The result is exact up to exponentially suppressed corrections governed by the pion mass times the box size. We also apply our master equation to various examples, including the two processes mentioned above as well as examples where the final state is an admixture of two open channels.

We use the halo model of clustering to compute two- and three-point correlation functions for weak lensing, and apply them in a new statistical technique to measure properties of massive haloes. We present analytical results on the eight shear three-point correlation functions constructed using a combination of the two shear components at each vertex of a triangle. We compare the amplitude and configuration dependence of the functions with ray-tracing simulations and find excellent agreement for different scales and models. These results are promising, since shear statistics are easier to measure than the convergence. In addition, the symmetry properties of the shear three-point functions provide a new and precise way of disentangling the lensing E mode from the B mode due to possible systematic errors. We develop an approach based on correlation functions to measure the properties of galaxy-group and cluster haloes from lensing surveys. Shear correlations on small scales arise from the lensing matter within haloes of mass M ? 10 1 3 M O .. Thus, the measurement of two- and three-point correlations can be used to extract information on halo density profiles, primarily the inner slope and halo concentration. We demonstrate the feasibility of such an analysis for forthcoming surveys. We include covariances in the correlation functions due to sample variance and intrinsic ellipticity noise to show that 10 per cent accuracy on profile parameters is achievable with surveys such as the CFHT Legacy survey, and significantly better with future surveys. Our statistical approach is complementary to the standard approach of identifying individual objects in survey data and measuring their properties. It can be extended to perform a combined analysis of the large-scale, perturbative regime down to small, subarcminute scales, to obtain consistent measurements of cosmological parameters and halo properties.

In this pedagogical lecture, I introduce some of the basic terminology and description of fluctuating fields as they occur on cosmology. I define various statistical, cosmological and sample homogeneity and explain what is meant by the fair sample hypothesis and cosmic variance. I illustrate these concepts using the simplest second-order statistics, i.e. the two-point correlation function and its Fourier transform the power-spectrum. I then give a brief overview of the properties of information relating to the properties of the phases of the Fourier modes of cosmological fluctuations which is not contained in these simpler statistics. Specifically, I explain how phase information of a particular form (called quadratic phase coupling) is encoded in the three-point correlation function (or, equivalently, the bispectrum).

The decays of charmed mesons into two body nonleptonic final states are investigated. Weak interaction amplitudes of interest in these decays are extracted from lattice four-point correlation functions using a effective weak Hamiltonian including effects to order Gf in the weak interactions yet containing effects to all orders in the strong interactions. The lattice calculation allows a quantitative examination of non-spectator processes in charm decays helping to elucidate the role of effects such as color coherence, final state interactions and the importance of the so called weak annihilation process. For D → Kπ, we find that the non-spectator weak annihilation diagram is not small, and we interpret this as evidence for large final state interactions. Moreover, there is indications of a resonance in the isospin 1/2 channel to which the weak annihilation process contributes exclusively. Findings from the lattice calculation are compared to results from the continuum vacuum saturation approximation and amplitudes are examined within the framework of the 1/N expansion. Factorization and the vacuum saturation approximation are tested for lattice amplitudes by comparing amplitudes extracted from lattice four-point functions with the same amplitude extracted from products of two-point and three-point lattice correlation functions arising out of factorization and vacuum saturation.

The evolution of density correlations in an expanding universe can be described by the BBGKY equations. This approach has been the subject of several previous studies, but always under the assumption of small-amplitude fluctuations, where the hierarchy of equations has a natural truncation. Reslts of these studies cannot be compared to the present universe because the galaxy two-point correlation function xi (r) is much greater than unity at r9 or approx. =1h/sup -1/ Mpc, and the three-point function zeta is on the order of xi (r)/sup 2/. In this strongly nonlinear situation the hierarchy is dominated by terms ignored in the linear analysis. Our method of truncating the hierarchy is based on the empirical result that zeta can be represented to good accuracy as a simple function of xi. We solve the equations via the velocity-moment method, and we truncate the resulting velocity-moment hierarchy for the two-point function by assuming that the distribution in the relative velocity of particle pairs has zero skewness about the mean. The second equation in this velocity-moment hierarchy is our main equation for xi. It involves the three-point spatial correlation function zeta, which we write as a function of xi following the empirical result. Themore » third equation involves the first velocity moment of the three-point position and velocity correlation function. We model this term in a way consistent with our model for zeta and with a constraint equation that expresses conservation of triplets.The equations admit a similarity transformation if (1) the effects of the discreteness of particles can be ignored, (2) the initial spectrum of density perturbations assumes a power law shape, and (3) the universe is described by an Einstein-de Sitter model (..cap omega..approx. =1). The numerical results presented here are based on this similarity solution.« less

We present an analytic study of the density fluctuation of a Newtonian self-gravity fluid in the expanding universe with ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{\ensuremath{\Lambda}}}+{\mathrm{\ensuremath{\Omega}}}_{m}=1$, which extends our previous work in the static case. By the use of field theory techniques, we obtain the nonlinear, hyperbolic equation of two-point correlation function $\ensuremath{\xi}$ of perturbation. Under the Zel'dolvich approximation the equation becomes an integro-differential equation and contains also the three-point and four-point correlation functions. By adopting the Groth-Peebles and Fry-Peebles ansatz, the equation becomes closed, and contains a pressure term and a delta source term which were neglected in Davis and Peebles' milestone work. The equation has three parameters of fluid; the particle mass $m$ in the source, the overdensity $\ensuremath{\gamma}$, and the sound speed ${c}_{s}$. We solve only the linear equation in linear approximation and apply it to the system of galaxies. We assume two models of ${c}_{s}$, and take an initial power spectrum at a redshift $z=7$, which inherits the relevant imprint from the spectrum of baryon acoustic oscillations at the decoupling. The solution $\ensuremath{\xi}(\mathbf{r},z)$ is growing during expansion, and contains 100 Mpc periodic bumps at large scales, and a main mountain (a global maximum with $\ensuremath{\xi}\ensuremath{\propto}{r}^{\ensuremath{-}1}$) at small scales $r\ensuremath{\lesssim}50\text{ }\text{ }\mathrm{Mpc}$. The profile of $\ensuremath{\xi}$ agrees with the observed ones from galaxy and quasar surveys. The bump separation is given by the Jeans length ${\ensuremath{\lambda}}_{J}$, and is also modified by $\ensuremath{\gamma}$ and ${c}_{s}$. Using a decomposition we find that the main mountain is largely generated by the inhomogeneous solution with the source, and the periodic bumps come from the homogeneous solution with the initial spectrum. ${\ensuremath{\lambda}}_{J}$ is identified as the correlation scale of the system of galaxies, distinguished from the clustering scale determined by $m$. The corresponding power spectrum has a main peak located around $k\ensuremath{\sim}\frac{2\ensuremath{\pi}}{{\ensuremath{\lambda}}_{J}}$ associated with the periodic bumps of $\ensuremath{\xi}$, and also contains multiwiggles at high $k&gt;{k}_{J}$ which are developing during evolution even if the initial spectrum has no wiggles. Since the outcome is affected by the initial condition and the parameters as well, it is hard to infer the imprint of baryon acoustic oscillations accurately. The difficulties with the sound horizon as a distance ruler are pointed out.

We extend previous work on the clustering properties of Gaussian random noise, and apply the results to cosmological density fields. We compute the three-point correlation function of regions where a Gaussian random process exceeds some threshold level. In the case where the threshold is sharp we find that the results are not well described by Kirkwood scaling; this contrasts with previously published work. As an alternative to the high-region prescription, we explore the clustering of local maxima of the density field. (Technical complexities involved with the treatment of local maxima restrict us to the simplified case of noise defined on a one-dimensional domain.) In this case, we find that the three-point correlation function follows a roughly hierarchical relationship with the two-point function in interesting scales

An investigation of the structure of the 0 + glueball in SU(2) lattice gauge theory