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

Shu-Guang Wu and Yang Zhang

Department of Astronomy,Key Laboratory for Researches in Galaxies and Cosmology,University of Science and Technology of China,Hefei 230026,China wusg@mail.ustc.edu.cn,yzh@ustc.edu.cn

Abstract 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.

Key words: gravitation–hydrodynamics–cosmology:large-scale structure of universe

1.Introduction

Then-point correlation functions are important tools to study the statistical properties of matter distribution on the large scale of the universe and can provide fundamental tests of the standard cosmological model(Peebles1980;Bernardeau et al.2002).The statistic of noninteracting particles,like CMB,can be well described a statistically Gaussian random field,the 2-point correlation function (2PCF) will be sufficient to characterize its correlation.When long-range Newtonian gravity is taken into account,the concept of a Gaussian random field has been subtle in literature so far.Therefore,a criterion of non-Gaussianity is required to be defined clearly.The equation of 2PCFG(2)(r) of density fluctuation to lowest order under Newtonian gravity is a Helmholtz equation with a delta source,and the exact solution has been given and called the solution in the Gaussian approximation in Zhang (2007).This is because the equationG(2)(r) shares a structure similar to the Gaussian approximate equation (Goldenfeld1992) that has been commonly used in condensed matter physics.Parallelly,the equation of 3-pt correlation function(3PCF) of density fluctuation to the lowest order (the Gaussian approximation) is also a linear equation and the exact solution(Zhang et al.2019) has been found as the following

whereQ=1,andr12=|r1-r2|,etc.Thus,Q=1 holds in the Gaussian approximation,and any deviation ofQfrom 1 will be an indication of non-Gaussianity of the density fluctuation.Interestingly,the solution(1)in the Gaussian approximation is exactly the content of the Groth-Peebles ansatz withQ=1(Groth&Peebles1975,1977).When density fluctuations up to second order are included,the equations ofG(2)(r) becomes nonlinear (Zhang &Miao2009;Zhang &Chen2015;Zhang et al.2019),and its solution describes the distribution of galaxies better than the Gaussian approximation at small scales.ButG(3)has not been analytically studied up to second order of density fluctuation.Statistically,G(3) (r ,r ′,r″)describes the excess probability over random of finding three galaxies located at the three vertices (r,r′,r″) of a given triangle.In observations and numerical studies,as an extension of the Groth-Peebles ansatz(1),the reduced 3PCF is often introduced

As a direct criterion,Q (r ,r ′,r″)indicates the non-Gaussianity when it deviates from 1.Galaxy surveys show thatQ≠1,and confirm the non-Gaussianity of the distribution of galaxies.Moreover,Qdepends on the scale and shape of the triangle(Jing&B?rner1998,2004;Wang et al.2004;Gazta?aga et al.2005;Nichol et al.2006;Gazta?aga et al.2009;Marín2011;McBride et al.2011a,2011b;Guo et al.2016;Slepian et al.2017),a feature also occurring in simulations(Fry et al.1993;Barriga&Gazta?aga2002;Gazta?aga &Scoccimarro2005) and in the study by perturbation theory(Fry1994;Bernardeau et al.2002).

In this paper,as a continuation of a series of study(Zhang2007;Zhang &Miao2009;Zhang &Chen2015;Zhang et al.2019),we shall derive analytically the nonlinear field equation ofG(3)up to second order of density fluctuation beyond Gaussian approximation,give the solutionG(3).As have been shown(Zhang&Li2021),the evolution effect of correlation function of galaxies is not drastic within a low redshift range(z=0.5～0.0),so for simplicity we study the nonevolution case and compare with observations (z=0.16～0.47) (Marín2011) in this preliminary work,and the evolution case will be given in future.

2.Nonlinear Field Equation of 3-point Correlation Function

Within the framework of Newtonian gravity,the distribution of galaxies and clusters in a static universe can be described by the density field ψ with the equation (Zhang2007;Zhang &Miao2009;Zhang &Chen2015;Zhang et al.2019)

The connectedn-point correlation function of δψ is

where δψ(r)=ψ(r)-〈ψ(r)〉 is the fluctuation field around the expectation value 〈ψ(r)〉.(See Goldenfeld1992;Zhang2007;Zhang&Miao2009;Zhang&Chen2015;Zhang et al.2019).To derive the field equation of the 3-point correlation function G(3)(r ,r ′,r″),we take the ensemble average of Equation(3)in the presence ofJ,and take the functional derivative of this equation twice with respect to the sourceJ,and setJ=0.In calculation,the second term in Equation (3) is approximated by

where the second order fluctuation (δψ)2is kept and higher order terms have been neglected.In this paper on the 3PCF to the second order of density perturbation,we work only up to the order(δψ)2,which is consistent with our previous works on the 2PCF to second order perturbation (Zhang &Miao2009;Zhang &Chen2015;Zhang et al.2019).The higher order(δψ)3terms in the expansion (7) are the third order of perturbation,and will be the subject of future study.By lengthy and straightforward calculations,using the definition (6),we obtain the field equation of G(3)(r ,r ′,r″)up to the second order of density fluctuation as the following

whereG(2)(0)≡G(2)(r,r) and ψ0≡〈ψ(r)〉J=0=1,and ?≡?rdenoting the gradient with respect to r through out the paper.When the higher order terms,such asG(2)G(3)andG(4),are dropped,Equation (8) reduces to that of the Gaussian approximation.(See Equation (28) in Zhang et al.(2019).)

Yet,Equation (8) is not closed forG(3),as it hierarchically contains the higher order 4-point correlation functionG(4)terms.To deal with it,we adopt the Fry–Peebles ansatz(Fry&Peebles1978) as the following

3.The Solution of 3PCF Equation

where the three variables are defined as

which are demonstrated in Figure1.

Figure 1.The configuration of the triangle of G( 3) (r ,r ′,r″)in the spherical coordinate.Here we take the azimuth angle φ=0,r″=0 as the origin,and the vector r′ -r ″along with the z-axis.

Then Equation (12) is written in spherical coordinates as

where ξ(r)≡G(2)(|r|),aris the radial component of the vector parameter a,

Equation (15) of ζ(r,u,θ) in spherical coordinates will be solved in actual computation.The ratiou=2 is often taken in simulations and presentations of observational data,so that ζ(r,u,θ)has only two variables.We also take this in the following.

To solve Equation (15) for ζ,we need the 2PCF ξ(r).For a coherent comparison with observation,we shall use the observed ξ(r) given in Figure 5 of Marín (2011).We plot Figure2(a) to show the observed ξ(r) (red with dots) from Marín (2011),and the nonlinear solution ξ(r) (blue) from Zhang&Chen(2015).We also plot the function A (r ,u,θ)of(16) in Figure2(b).

Besides,we also need an appropriate boundary condition on some domain.Marín (2011) has obtained the redshift-space 3PCF of luminous red galaxies of“DR7-Dim”(61,899 galaxies in the range 0.16 ≤z≤0.36)from SDSS.In Figures 6 and 7 of Marín (2011),the reducedQ(s,u,θ) are given in the domains∈[7.0,30.0]h-1Mpc,θ ∈[0.1,3.04]at five respective valuess=7,10,15,20,30h-1Mpc at a fixedu=2.Specifically,we shall use the measuredQ(s,u,θ) ats=7h-1Mpc ands=30h-1Mpc as a part of the boundary condition,which is fitted by

Also from Figures 6 and 7 of Marín(2011),we give the fittedQ(s,u,θ)at θ=0.1 and θ=3.04 as another part of the boundary condition

(17) and (18) lead to the boundary values of ζ(s,u,θ)on the domain,by virtue of the relation (2).The redshift distancesis used in Marín (2011) which may differ from the real distancerdue to the peculiar velocities.We shall neglect this error in our computation.To match the observational data (Marín2011),the parameters are chosen as the following:ar=-1043.8hMpc-1,b=-1627.3,c=-36.4h2Mpc-2,g=-5586.6,Ra=1.66,Rb=-0.34,Q=1.1,kJ=0.161hMpc-1.

Equation (15) is a convection–diffusion partial differential equation,and we employ the streamline diffusion method (Elman et al.2014) to solve it numerically.We obtain the solution ζ(r,u,θ) and the reducedQ(r,u,θ) by the relation (2).

Figure3(a) plots the surface of ζ(r,u,θ) as a function of (r,θ),which exhibits a shallowU-shape along θ and turns up at θ ?π/2.This feature of solution is consistent with observations (Guo et al.2014,2016).ζ(r,u,θ) decreases monotonously alongrup to 30h-1Mpc.The highest values of ζ(r,u,θ) occur at smallrand θ.For a comparison,Figure3(b) plots the Gaussian solution ζg(r,u,θ) of Equation (1),which decreases monotonously along both θ andr,having noU-shape along θ.

Figure4plots the surface of reducedQ(r,u,θ)as a function of (r,θ),which deviates from the Gaussianity planeQ(r,u,θ)=1,exhibits a deeperU-shape along θ,and varies along the radialr.The highest values ofQ(r,u,θ)occur at largerand θ,just opposite to ζ(r,u,θ).The variation alongris comparatively weaker than the variation along θ.These features are consistent with observations (Marín2011;McBride et al.2011a,2011b).

Figure 2.(a):the observed ξ(r)(red with dots)from Marín(2011),the solution ξ to second order(blue)from Zhang&Chen(2015).(b): A (r ,u,θ)in Equation(16)at fixed u=2 as function of (r,θ).

Figure 3.(a):The solution ζ(r,u,θ)shows a shallower U-shape along θ,and decreases monotonously along r.(b):The Gaussian solution ζg(r,u,θ)of Equation(1)decreases monotonously along both θ and r.

Figure 5.The solid line:Q(r,u,θ) at u=2 converted from the solution ζ(r,u,θ).The points:the SDSS observational data from Figures 6 and 7 of Marín (2011).Three plots are for r=10h-1Mpc,15h-1Mpc,20h-1Mpc,respectively.Q(r,u,θ)deviates from Q(r,u,θ)=1 of Gausianity and forms a U-shape along the elevation angle θ ∈[0,3],agreeing with the data.

To compare with observations,Figure5showsQ(r,u,θ)as a function of θ at respectively fixedr=10,15,20h-1Mpc.Q(r,u,θ)agrees well with the data of Marín(2011)available in the range θ=(0.1～3.0).

Figure 6.Similar to Figure5.Q(r,u,θ)isplotted,using another set of parameters:kJ=0.12822 h Mpc-1,ar=34.03 kJ,b=3.36,c=1.8844 h2Mpc-2,Ra=-2.06,Rb=6.64,Q=0.7 with g=1+c ().The f i tting to the data is not as good as Figure 5.

As an example,Figure6plotsQ(r,u,θ) with another set of parameter values,and the f i tting is not as good as that in Figure5.

4.Conclusions and Discussions

We have presented an analytical study on the 3-point correlation function of galaxies based on the f i eld theory of density f l uctuations of a Newtonian gravitating system,and have derived the nonlinear f i eld Equation (8) ofG(3)up to the second order density f l uctuation.This work is a continuation of the previous works on the 2PCF (Zhang2007;Zhang &Miao2009;Zhang &Chen2015) and on the Gaussian 3PCF(Zhang et al.2019).

By adopting the Fry–Peebles ansatz to deal with the 4PCF,and the Groth-Peebles ansatz to deal with the squeezed 3PCF,respectively,we have made Equation (8) into the closed Equation (12) ofG(3),equivalently Equation (15) of ζ in spherical coordinates.For coherency,we have used the observed 2PCF and the boundary condition from SDSS DR7(Marín2011),in solving for the 3PCF.

The solution ζ(r,u,θ) exhibits a shallowU-shape along θ,agreeing with the observed one.Nevertheless,ζ(r,u,θ)decreases monotonously alongr,at least up to 30h-1Mpc of the domain in our computation.For comparison,we also plot the Gaussian solution ζg(r,u,θ),which decreases monotonously along both θ andr,having noU-shape along θ.The difference between ζ and ζgimplies the non-Gaussianity of the distribution of galaxies.

The non-Gaussianity is directly indicated by the reducedQ(r,u,θ).The solutionQ(r,u,θ)deviates from the Gaussianity planeQ(r,u,θ)=1,also exhibits aU-shape along θ,just like ζ(r,u,θ),agreeing with the observations (Marín2011).In fact,by its de f i nition(2),Q(r,u,θ)shares the same θ-dependence as ζ(r,u,θ),and its denominator consists of θ-independent ξ(r).Alongr,however,Q(r,u,θ)varies non-monotonically,scattering around 1,unlike ζ(r,u,θ).Moreover,the highest values ofQ(r,u,θ)occur at largerand θ,a behavior just opposite to ζ(r,u,θ).These two features ofQ(r,u,θ) are due to the behavior of ξ(r)which is large at smallrand suppressesQ(r,u,θ) thereby.

This preliminary study of 3PCF in this paper should be extended,and several issues need more investigation in future,such as the impact of physical parameters,exploration of parameter space in association with 2PCF,and the effect of cosmic expansion (Zhang &Li2021).

Acknowledgments

Y.Zhang is supported by NSFC Grant Nos.11675165,11633001 and 11961131007 and in part by the National Key RD Program of China (2021YFC2203100).