Three-point Correlation Research Articles

Towards a self-consistent analysis of the anisotropic galaxy two- and three-point correlation functions on large scales: application to mock galaxy catalogues

ABSTRACT We establish a practical method for the joint analysis of anisotropic galaxy two- and three-point correlation functions (2PCF and 3PCF, respectively) on the basis of the decomposition formalism of the 3PCF using tripolar spherical harmonics. We perform such an analysis with MultiDark-Patchy mock catalogues to demonstrate and understand the benefit of the anisotropic 3PCF. We focus on scales above $80\, h^{-1}\, {\rm Mpc}$, and use information from the shape and the baryon acoustic oscillation (BAO) signals of the 2PCF and 3PCF. We also apply density field reconstruction to increase the signal-to-noise ratio of BAO in the 2PCF measurement, but not in the 3PCF measurement. In particular, we study in detail the constraints on the angular diameter distance and the Hubble parameter. We build a model of the bispectrum or 3PCF that includes the non-linear damping of the BAO signal in redshift space. We carefully account for various uncertainties in our analysis including theoretical models of the 3PCF, window function corrections, biases in estimated parameters from the fiducial values, the number of mock realizations to estimate the covariance matrix, and bin size. The joint analysis of the 2PCF and 3PCF monopole and quadrupole components shows a $30{{\ \rm per\ cent}}$ and $20{{\ \rm per\ cent}}$ improvement in Hubble parameter constraints before and after reconstruction of the 2PCF measurements, respectively, compared to the 2PCF analysis alone. This study clearly shows that the anisotropic 3PCF increases cosmological information from galaxy surveys and encourages further development of the modelling of the 3PCF on smaller scales than we consider.

The Catalogue for Astrophysical Turbulence Simulations (CATS)

Abstract Turbulence is a key process in many fields of astrophysics. Advances in numerical simulations of fluids over the last several decades have revolutionized our understanding of turbulence and related processes such as star formation and cosmic ray propagation. However, data from numerical simulations of astrophysical turbulence are often not made public. We introduce a new simulation-oriented database for the astronomical community: the Catalogue for Astrophysical Turbulence Simulations (CATS), located at www.mhdturbulence.com. CATS includes magnetohydrodynamic (MHD) turbulent box simulation data products generated by the public codes athena++, arepo, enzo, and flash. CATS also includes several synthetic observational data sets, such as turbulent HI data cubes. We also include measured power spectra and three-point correlation functions from some of these data. We discuss the importance of open-source statistical and visualization tools for the analysis of turbulence simulations such as those found in CATS.

The semi-leptonic form factors of Lambda _{b}rightarrow Lambda _{c} and Xi _{b}rightarrow Xi _{c} in QCD sum rules

In this work, the full leading order results of the form factors for Xi _{b}rightarrow Xi _{c} and Lambda _{b}rightarrow Lambda _{c} are obtained in QCD sum rules. Contributions from up to dimension-5 have been considered. For completeness, we also study the two-point correlation function to obtain the pole residues of Xi _{Q} and Lambda _{Q}, and higher accuracy is achieved. For the three-point correlation function, since stable Borel regions cannot be found, about 20% uncertainties are introduced for the form factors of Xi _{b}rightarrow Xi _{c} and Lambda _{b}rightarrow Lambda _{c}. Our results for the form factors are consistent with those of the lattice QCD within errors.

OPE of Green functions of chiral currents

In this paper, we investigate the high-energy behavior of two-point and three-point Green functions of the QCD chiral currents and densities using the framework of the operator product expansion in the chiral limit. In detail, we study the contributions of the quark, gluon, quark-gluon and four-quark condensates to all the relevant non-vanishing three-point correlators.

Three- and two-point spatial correlations of IGM at z ∼ 2: cloud-based analysis using simulations

ABSTRACT Ly α forest absorption spectra decomposed into multiple Voigt profile components (clouds) allow us to study clustering of intergalactic medium (IGM) as a function of H i column density ($N_{\rm H\, \small{I}}$). Here, we explore the transverse three-point correlation (ζ) of Ly α clouds using mock triplet spectra from simulations at z ∼ 2 on scales of 1–5 h−1 cMpc. We find ζ to depend strongly on $N_{\rm H\, \small{I}}$ and scale and weakly on angle of the triplet configuration. We show that the ‘hierarchical ansatz’ is applicable for scales $\ge ~ 3\, h^{-1}$ cMpc, and obtain a median reduced three-point correlation (Q) in the range 0.2–0.7. We show, ζ is influenced strongly by the thermal and ionization state of the gas. As in the case of galaxies, the influence of physical parameters on Q is weaker compared to that on ζ. We show difference in ζ and Q between different simulations are minimized if we use appropriate $N_{\rm H\, \small{I}}$ cut-offs corresponding to a given baryon overdensity (Δ) using the $N_{\rm H\, \small{I}}~versus~\Delta$ relationship obtained from simulations. We study the effect of pressure broadening on ζ using a model with artificially boosted heating rates. However, for models with realistic thermal and ionization histories the effect of pressure broadening on ζ is subdominant compared to other local effects. We find the strong redshift evolution shown by ζ, mainly originating from the redshift evolution of thermal and ionization state of the IGM. We discuss the observational requirements for the detection of three-point correlation, specifically, in small intervals of configuration parameters and redshift.

Analysing the Epoch of Reionization with three-point correlation functions and machine learning techniques

ABSTRACT Three-point and high-order clustering statistics of the high-redshift 21 cm signal contain valuable information about the Epoch of Reionization (EoR). We present 3PCF-fast, an optimized code for estimating the three-point correlation function (3PCF) of 3D pixelized data such as the outputs from numerical and seminumerical simulations. After testing 3PCF-fast on data with known analytical 3PCF, we use machine learning techniques to recover the mean bubble size and global ionization fraction from correlations in the outputs of the publicly available 21cmfast code. We assume that foregrounds have been perfectly removed and negligible instrumental noise. Using ionization fraction data, our best multilayer perceptron (MLP) model recovers the mean bubble size with a median prediction error of around $10 {{\ \rm per\ cent}}$, or from the 21 cm differential brightness temperature with median prediction error of around $14 {{\ \rm per\ cent}}$. A further two MLP models recover the global ionization fraction with median prediction errors of around $4 {{\ \rm per\ cent}}$ (using ionization fraction data) or around $16 {{\ \rm per\ cent}}$ (using brightness temperature). Our results indicate that clustering in both the ionization fraction field and the brightness temperature field encode useful information about the progress of the EoR in a complementary way to other summary statistics. Using clustering would be particularly useful in regimes where high signal-to-noise ratio prevents direct measurement of bubble size statistics. We compare the quality of MLP models using the power spectrum, and find that using the 3PCF outperforms the power spectrum at predicting both global ionization fraction and mean bubble size.

Vertex algebraic intertwining operators among generalized Verma modules for affine Lie algebras

We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra gˆ, from g-module homomorphisms. When g=sl2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to gˆ, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.

KiDS+VIKING+GAMA: Testing semi-analytic models of galaxy evolution with galaxy–galaxy–galaxy lensing

Context. Several semi-analytic models (SAMs) try to explain how galaxies form, evolve, and interact inside the dark matter large-scale structure. These SAMs can be tested by comparing their predictions for galaxy–galaxy–galaxy lensing (G3L), which is weak gravitational lensing around galaxy pairs, with observations. Aims. We evaluate the SAMs by Henriques et al. (2015, MNRAS, 451, 2663, hereafter H15) and by Lagos et al. (2012, MNRAS, 426, 2142, hereafter L12), which were implemented in the Millennium Run, by comparing their predictions for G3L to observations at smaller scales than previous studies and also for pairs of lens galaxies from different populations. Methods. We compared the G3L signal predicted by the SAMs to measurements in the overlap of the Galaxy And Mass Assembly survey (GAMA), the Kilo-Degree Survey (KiDS), and the VISTA Kilo-degree Infrared Galaxy survey (VIKING) by splitting lens galaxies into two colour and five stellar-mass samples. Using an improved G3L estimator, we measured the three-point correlation of the matter distribution with “mixed lens pairs” with galaxies from different samples, and with “unmixed lens pairs” with galaxies from the same sample. Results. Predictions by the H15 SAM for the G3L signal agree with the observations for all colour-selected samples and all but one stellar-mass-selected sample with 95% confidence. Deviations occur for lenses with stellar masses below 9.5 h−2 M⊙ at scales below 0.2 h−1 Mpc. Predictions by the L12 SAM for stellar-mass selected samples and red galaxies are significantly higher than observed, while the predicted signal for blue galaxy pairs is too low. Conclusions. The L12 SAM predicts more pairs of low stellar mass and red galaxies than the H15 SAM and the observations, as well as fewer pairs of blue galaxies. This difference increases towards the centre of the galaxies’ host halos. Likely explanations are different treatments of environmental effects by the SAMs and different models of the initial mass function. We conclude that G3L provides a stringent test for models of galaxy formation and evolution.

Electromagnetic coupling effects in natural inflation

In this work we study the effects of the electromagnetic coupling in natural inflation in a systematic manner using the Schwinger–Keldysh formalism. The corresponding influence functional is evaluated in the one-loop level. It can be interpreted as due to a single stochastic force. The equation of motion of the inflaton field is therefore given in the form of a Langevin equation. Lastly, the two-point and the three-point correlation functions of the inflaton field are worked out. They are related to the power spectrum and the nongaussianity of the inflaton field, respectively.

Conduction in 2-D and 3-D dimensional spherically-symmetric anisotropic-coating inclusion composites

Based on the variational three-point correlation results for the conductivity (thermal, electrical,...) of the multi-coated sphere assemblages, limiting procedures have been developed to construct the explicit expressions of the macroscopic conductivity and the respective microscopic gradient and flux fields for spherically-symmetric inclusion composites with anisotropic coating in d dimensions (d=2,3) under both the imposed macroscopic gradient or flux fields. Numerical examples are provided, verifying the convergence of the results for the macroscopic conductivity obtained from the minimum energy upper and lower bound approaches, and also that derived from the differential one. Explicit formal and numerical expressions of the microscopic conduction fields and illustration pictures of the microscopic flux lines within and around the typical anisotropic coating inclusion are also given. When the volume proportion of the outermost spherical shell increases to be the predominant one, one obtains the respective exact results for the important specific case of dilute solution of the anisotropic-coating inclusions suspended in the major matrix phase.

Bond percolation between k separated points on a square lattice.

We consider a percolation process in which k points separated by a distance proportional the system size L simultaneously connect together (k>1), or a single point at the center of a system connects to the boundary (k=1), through adjacent connected points of a single cluster. These processes yield new thresholds p[over ¯]_{ck} defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of p_{ck} for individual samples remains broad in the limit of L→∞. We study p[over ¯]_{ck} for bond percolation on the square lattice and find that p[over ¯]_{ck} are above the normal percolation threshold p_{c}=1/2 and represent specific supercritical states. The p[over ¯]_{ck} can be related to integrals over powers of the function P_{∞}(p) equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P_{∞}(p) on L×L systems that for L→∞,p[over ¯]_{c1}=0.51755(5),p[over ¯]_{c2}=0.53219(5),p[over ¯]_{c3}=0.54456(5), and p[over ¯]_{c4}=0.55527(5). The percolation thresholds p[over ¯]_{ck} remain the same, even when the k points are randomly selected within the lattice. We show that the finite-size corrections scale as L^{-1/ν_{k}} where ν_{k}=ν/(kβ+1), with β=5/36 and ν=4/3 being the ordinary percolation critical exponents, so that ν_{1}=48/41,ν_{2}=24/23,ν_{3}=16/17,ν_{4}=6/7, etc. We also study three-point correlations in the system and show how for p>p_{c}, the correlation ratio goes to 1 (no net correlation) as L→∞, while at p_{c} it reaches the known value of 1.022.

Dynamics and invariants of the perceived velocity gradient tensor in homogeneous and isotropic turbulence

The perceived velocity gradient tensor (PVGT), constructed from four fluid tracers forming a tetrahedron, provides a natural way to study the structure of velocity fluctuations and its dependence on spatial scales. It generalizes and shares qualitatively many properties with the true velocity gradient tensor. Here, we establish the evolution equation for the PVGT, and, for homogeneous and isotropic incompressible turbulent flows, we analyse the dynamics of the PVGT in particular using its second- and third-order invariants. We show that, for PVGT based on regular tetrads with lateral size , the second-order invariants can be expressed solely in terms of the usual second-order velocity structure functions, while the third-order invariants involve the usual third-order longitudinal velocity structure function and a less well known three-point velocity correlation function. For homogeneous and isotropic turbulence, exact relations between the second moments of strain and vorticity, as well as enstrophy production and the third moments of the strain, are derived. These generalized relations are valid for all ranges of , and reduce to classical results for the velocity gradient tensor when is in the dissipative range. With the help of these relations, we quantify the importance of the various terms, such as vortex stretching, as a function of the scale . Our analysis, which is supported by the results of direct numerical simulations of turbulent flows in the Reynolds-number range , allows us to demonstrate that strain prevails over vorticity when is in the inertial range.

A QCD Sum Rules Calculation of the $J/\psi D_{s}^{*} D_{s}^{*}$ Form Factors and Strong Coupling Constants

We use the QCD sum rules for the three-point correlation functions to compute the strong coupling constants of the meson vertices $J/\psi D_{s}^{*} D_{s}^{*}$. Using the QCD sum rules, we obtain the form factor of the vertex, the coupling constant, and the cut off parameter. Uncertainties are included. The results obtained for the coupling constants are $g_{J/\psi D^{*}_{s} D^{*}_{s}} = 7.47^{+1.04}_{-0.71}.$.

Detecting the Number of States in Raw Trajectories

In this paper, we present a simple method to detect the number of states in a stochastic trajectory. The method quantifies the degree of correlations in stochastic trajectories, computes the correlation function with two variables (the three-point correlation function), then finds the rank of the computed matrix (the method identifies the signal singular values, those that are beyond the noise). The computed rank is the number of states in the discrete trajectory, yet meaningful also in continuous trajectories; in such cases, the rank is compiled with the number of terms in the correlation function to determine the number of fluctuating independent potential profiles in the approximated discrete representation of the process.

The weak lensing bispectrum induced by gravity

ABSTRACT Recent studies have demonstrated that secondary non-Gaussianity induced by gravity will be detected with a high signal-to-noise ratio (S/N) by future and even by on-going weak lensing surveys. One way to characterize such non-Gaussianity is through the detection of a non-zero three-point correlation function of the lensing convergence field, or of its harmonic transform, the bispectrum. A recent study analysed the properties of the squeezed configuration of the bispectrum, when two wavenumbers are much larger than the third one. We extend this work by estimating the amplitude of the (reduced) bispectrum in four generic configurations, i.e. squeezed, equilateral, isosceles and folded, and for four different source redshifts zs = 0.5, 1.0, 1.5, 2.0, by using an ensemble of all-sky high-resolution simulations. We compare these results against theoretical predictions. We find that, while the theoretical expectations based on widely used fitting functions can predict the general trends of the reduced bispectra, a more accurate theoretical modelling will be required to analyse the next generation of all-sky weak lensing surveys. The disagreement is particularly pronounced in the squeezed limit.

Bootstrapping inflationary correlators in Mellin space

We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdSd+1 and dSd+1 is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dSd+1 from known results for Witten diagrams in AdSd+1 by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdSd+1 and dSd+1 in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin-ℓ. The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general d and generic scaling dimensions, in particular massive, light and (partially-)massless fields. It moreover naturally identifies boundary correlation functions for generic fields with multi-variable Meijer-G functions. When d = 3 we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin- ℓ field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.

Measuring galaxy-galaxy-galaxy-lensing with higher precision and accuracy

Context. Galaxy-galaxy-galaxy lensing (G3L) is a powerful tool for constraining the three-point correlation between the galaxy and matter distribution and thereby models of galaxy evolution. Aims. We propose three improvements to current measurements of G3L: (i) a weighting of lens galaxies according to their redshift difference, (ii) adaptive binning of the three-point correlation function, and (iii) accounting for the effect of lens magnification by the cosmic large-scale structure. Improvement (i) is designed to improve the precision of the G3L measurement, whereas improvements (ii) and (iii) remove biases of the estimator. We further show how the G3L signal can be converted from angular into physical scales. Methods. The improvements were tested on simple mock data and simulated data based on the Millennium Run with an implemented semi-analytic galaxy model. Results. Our improvements increase the signal-to-noise ratio by 35% on average at angular scales between 0.′1 and 10′ and physical scales between 0.02 and 2 h−1 Mpc. They also remove the bias of the G3L estimator at angular scales below 1′, which was originally up to 40%. The signal due to lens magnification is approximately 10% of the total signal.

Anomalous gravitational TTT vertex, temperature inhomogeneity, and pressure anisotropy

The conformal anomaly in curved spacetime generates a nontrivial anomalous vertex, given by the three-point correlation function TTT of the energy-momentum tensor Tμν. We show that a temperature inhomogeneity in a gas of charged massless particles generates, via the TTT vertex, a pressure anisotropy with respect to the axis of the temperature variation. This very particular signature may provide an experimental access to the elusive gravitational coefficient b which determines the anomaly contribution of the Weyl tensor to the trace of the energy-momentum tensor in curved spacetime. We present an estimate of the pressure anisotropy both for fermionic quasiparticles in the solid-state environment of Dirac semimetals as well as for a quark-gluon plasma in relativistic heavy-ion collisions. In both cases, the pressure anisotropy is small compared to the mean thermal pressure.

Baryon acoustic oscillations signature in the three-point angular correlation function from the SDSS-DR12 quasar survey

ABSTRACTThe clustering properties of the Universe at large scales are currently being probed at various redshifts through several cosmological tracers and with diverse statistical estimators. Here we use the three-point angular correlation function (3PACF) to probe the baryon acoustic oscillation (BAO) features in the quasars catalogue from the Sloan Digital Sky Survey Data Release 12, with mean redshift $\overline{z} = 2.225$, detecting the BAO imprint with a statistical significance of $2.9 \sigma$, obtained using lognormal mocks. Following a quasi-model-independent approach for the 3PACF, we find the BAO transversal signature for triangles with sides θ1 = $1{^{\circ}_{.}}0$ and θ2 = $1{^{\circ}_{.}}5$ and the angle between them of α = 1.59 ± 0.17 rad, a value that corresponds to the angular BAO scale $\theta_{\rm BAO}=1{^{\circ}_{.}}82 \pm 0{^{\circ}_{.}}21$, in excellent agreement with the value found in a recent work ($\theta_{\rm BAO}=1{^{\circ}_{.}}77 \pm 0{^{\circ}_{.}}31$) applying the two-point angular correlation function (2PACF) to similar data. Moreover, we performed two types of test: one to confirm the robustness of the BAO signal in the 3PACF through random displacements in the data set, and the other to verify the suitability of our random samples, a null test that in fact does not show any signature that could bias our results.

Lorentzian CFT 3-point functions in momentum space

In a conformal field theory, two and three-point functions of scalar operators and conserved currents are completely determined, up to constants, by conformal invariance. The expressions for these correlators in Euclidean signature are long known in position space, and were fully worked out in recent years in momentum space. In Lorentzian signature, the position-space correlators simply follow from the Euclidean ones by means of the iϵ prescription. In this paper, we compute the Lorentzian correlators in momentum space and in arbitrary dimensions for three scalar operators by means of a formal Wick rotation. We explain how tensorial three-point correlators can be obtained and, in particular, compute the correlator with two identical scalars and one energy-momentum tensor. As an application, we show that expectation values of the ANEC operator simplify in this approach.