Research Note
Field equation of the correlation function of massdensity fluctuations for selfgravitating systems
Department of Astronomy, Key Laboratory for Researches in Galaxies and Cosmology, University of Science and Technology of China, Hefei, Anhui 230026, PR China
email: yzh@ustc.edu.cn; cqpb@mail.ustc.edu.cn
Received: 28 November 2014
Accepted: 1 July 2015
We study the massdensity distribution of Newtonian selfgravitating systems. Modeling the system as a fluid in hydrostatical equilibrium, we obtain from first principles the field equation and its solution of the correlation function ξ(r) of the massdensity fluctuation itself. We apply this to studies of the largescale structure of the Universe within a small redshift range. The equation shows that ξ(r) depends on the point mass m and the Jeans wavelength scale λ_{0}, which are different for galaxies and clusters. It explains several longstanding prominent features of the observed clustering: that the profile of ξ_{cc}(r) of clusters is similar to ξ_{gg}(r) of galaxies, but with a higher amplitude and a longer correlation length, and that the correlation length increases with the mean separation between clusters as a universal scaling r_{0} ≃ 0.4d. Our solution ξ(r) also shows that the observed powerlaw correlation function of galaxies ξ_{gg}(r) ≃ (r_{0}/r)^{1.7} is only valid in a range 1 <r< 10h^{1} Mpc. At larger scales the solution ξ(r) breaks below the power law and goes to zero around ~50 h^{1} Mpc, just as observational data have demonstrated. With a set of fixed model parameters, the solutions ξ_{gg}(r) for galaxies, the corresponding power spectrum, and ξ_{cc}(r) for clusters, simultaneously agree with the observational data from the major surveys of galaxies and clusters.
Key words: galaxies: clusters: general / largescale structure of Universe / gravitation / cosmology: theory
© ESO, 2015
1. Introduction
It is one of the major goals of modern cosmology to understand the matter distribution in the Universe on large scales. The largescale structure is determined by the selfgravity of galaxies and clusters. Since the number of galaxies is enormous, one needs statistics to study their distribution. In this regard, the twopoint correlation functions ξ_{gg}(r) of galaxies and ξ_{cc}(r) clusters serve as a powerful statistical tool (Bok 1934; Totsuji & Kihara 1969; Peebles 1980). They not only provide statistical information, but also contain underlying dynamics that are mainly due to gravitational force. Therefore, we would like to investigate the correlation functions of selfgravitating systems in an approximation of hydrostatical equilibrium.
Over the years, various observational surveys have been carried out for galaxies and clusters, such as the Automatic Plate Measuring (APM) galaxy survey (Loveday et al. 1996), the TwodegreeField Galaxy Redshift Survey (2dFGRS; Peacock et al. 2001), and the Sloan Digital Sky Survey (SDSS; Abazajian et al. 2009). All these surveys suggest that the correlation of galaxies has a powerlaw form ξ_{gg}(r) ∝ (r_{0}/r)^{γ} with r_{0} ~ 5.4h^{1} Mpc and γ ~ 1.7 in a range (0.1 ~ 10)h^{1} Mpc (Totsuji & Kihara 1969; Groth & Peebles 1977; Peebles 1980; Soneira & Peebles 1987). The correlation of clusters is found to be of a similar form: ξ_{cc}(r) ~ 20ξ_{gg}(r) in a range (5 ~ 60)h^{1} Mpc, with an amplified magnitude (Bahcall & Soneira 1983; Klypin & Kopylov 1983). For quasars, the correlation is ξ_{qq}(r) ~ 5ξ_{gg}(r) (Shaver 1988). Numerical computations have been extensively employed to study clustering of galaxies and clusters, and significant progresses have been made. To understand physical mechanisms behind the clustering, analytical studies are important.
In particular, Saslaw (1985, 2000) used thermodynamics whereby the powerlaw form of ξ_{gg}(r) was introduced as modifications to the energy and pressure. Similarly, de Vega et al. (1996a,b, 1998) used the grand partition function of a selfgravitating gas to study a possible fractal structure of the distribution of galaxies. However, the field equation of ξ was not given in these studies. In this paper we set the speed of light to c = 1 and the Boltzmann constant to k_{B} = 1.
2. Field equation of the twopoint correlation function of density fluctuations
Galaxies or clusters that are distributed in the Universe can be described as fluids at rest in gravitational fields. This modeling is an approximation since cosmic expansion is not considered. We apply hydrostatics to systems of galaxies within a small redshift range. For these selfgravitating systems, the field equation of mass density is (Zhang 2007; Zhang & Miao 2009) (1)where ψ(r) ≡ ρ(r) /ρ_{0} with ρ_{0} = mn_{0} is the mean mass density of the system, is the Jeans wave number, and J is a Schwinger type of external source introduced for taking the functional derivative (Schwinger 1951). The effective Hamiltonian density is (2)The generating functional for the correlation functions of ψ is (3)where , c_{s} is the sound speed and m is the mass of a single particle.
Since the distribution of galaxies or clusters can be viewed as fluctuations of the mass density in the Universe, we consider the fluctuation field δψ(r) ≡ ψ(r) − ⟨ ψ(r) ⟩, where the statistical ensemble average is defined as , and, in our case, ⟨ ψ(r) ⟩ = ψ_{0} is a constant. The connected npoint correlation function of δψ is defined as (Binney et al. 1992) (4)for n ≥ 2. One can take G^{(2)}(r_{1},r_{2}) = G^{(2)}(r_{12}) for the homogeneous and isotropic Universe. To derive the field equation of G^{(2)}(r) (Goldenfeld 1992), one takes functional derivative of the ensemble average of Eq. (1) with respect to J(r_{1}). In doing this, G^{(3)} occurs in the equation of G^{(2)}(r) hierarchically. We adopt the KirkwoodGrothPeebles ansatz (Kirkwood 1932; Groth & Peebles 1977) G^{(3)}(r_{1},r_{2},r_{3}) = Q [ G^{(2)}(r_{12})G^{(2)}(r_{23}) + G^{(2)}(r_{23})G^{(2)}(r_{31}) + G^{(2)}(r_{31})G^{(2)}(r_{12}) ] , where Q is a dimensionless parameter. Then, after a necessary renormalization, we obtain the field equation of the twopoint correlation function (5)where ξ = ξ(r) ≡ G^{(2)}(r), , x ≡ k_{0}r, , and a, b, and c are three independent parameters. Equation (5) extends that in our earlier work (Zhang & Miao 2009). The special case of a = b = c = 0 is the Gaussian approximation. The terms containing a, b, and c represent the nonlinear contributions beyond the Gaussian approximation. The nonlinear terms with b and c in Eq. (5) can enhance the amplitude of ξ at small scales and increase the correlation length. The term containing a plays the role of effective viscosity. The value of a should be high enough to ensure 1 + ξ(r) ≥ 0 for the whole range 0 <r< ∞.
3. General predictions of field equation
We inspect Eq. (5) to see its predictions on general properties of the correlation function ξ(r).
First, the equation contains a point mass m and a characteristic wave number k_{0}. It applies to the system of galaxies and to the system of clusters, but with different m and k_{0} in each respective case. Thus, as solutions of Eq. (5), ξ_{cc} for clusters should have a profile similar to ξ_{gg} for galaxies, but will differ in amplitude and in scale determined by different m and k_{0}. Indeed, the observations show that both ξ_{gg} and ξ_{cc} have a powerlaw form: ∝r^{1.8} in their respective finite range, but ξ_{cc} has a higher amplitude (Bahcall & Soneira 1983; Klypin & Kopylov 1983).
Second, the δ^{3}(r) source in Eq. (5) has a coefficient , which determines the overall amplitude of a solution ξ. The mass m of a cluster can be 10 ~ 10^{3} times that of a galaxy (Bahcall 1999), while c_{s} regarded as the peculiar velocity is around several hundreds km s^{1} for galaxies and clusters. Therefore, 1 /α ∝ m, and a greater m will yield a higher amplitude of ξ. This general prediction naturally explains a whole chain of prominent facts of observations: that luminous galaxies are more massive and have a higher correlation amplitude than ordinary galaxies (Zehavi et al. 2005), that clusters are much more massive and have a much higher correlation than galaxies, and that rich clusters have a higher correlation than poor clusters since richness is proportional to mass (Bahcall & Soneira 1983; Einasto et al. 2002, 2007; Bahcall et al. 2003). These phenomena have been a puzzle for long (Bahcall 1999) and were interpreted as being caused by the statistics of rare peak events (Kaiser 1984).
Third, the power spectrum is proportional to the inverse of the spatial number density: P(k) ∝ 1 /n_{0}. Given the mean mass density ρ_{0} = mn_{0}, a greater m implies a lower n_{0}. Therefore, properties 2 and 3 reflect the same physical law of clustering from different perspectives. Property 3 also agrees with the observed fact from a variety of surveys. The observed P(k) of clusters is much higher than that of galaxies, and the observed P(k) of rich clusters is higher than poor clusters, etc. This is because the n_{0} of clusters is much lower than that of galaxies, and n_{0} of rich clusters is lower than that of poor clusters (Bahcall 1999).
Fourth, the characteristic length appears in Eq. (5) as the only scale that underlies the scalerelated features of clustering. Cluster surveys extend over larger spatial volumes, including those of highly diluted regions. The mass density ρ_{0c} of the region covered by cluster surveys is lower than ρ_{0g} for galaxy surveys, as implied in the references (Bahcall 1999; Bahcall et al. 1995). Thus, λ_{0} for cluster surveys will be longer than that for galaxy surveys. As we show in Sects. 4 and 5, in using one solution ξ(r) to match the data of both galaxies and clusters, one needs to take k_{0} smaller for clusters than for galaxies.
4. Comparing observational data of galaxy surveys
Now we give the solution ξ_{gg}(r) for a fixed set of parameters (a,b,c) and compare the observed correlation from major galaxy surveys.
First we consider the correlation function ξ_{gg}(r).
Figure 1 shows the solution ξ_{gg}(r) and the observational data by the galaxy surveys of APM (Padilla & Baugh 2003), SDSS (Zehavi et al. 2005), and 2dFGRS (Hawkins et al. 2003). The theoretical ξ_{gg}(r) matches the data in the range of r = (2 ~ 50)h^{1} Mpc. The usual powerlaw ξ_{gg} ∝ r^{1.7} is only valid in an interval (0.1 ~ 10)h^{1} Mpc. On large scales, the solution ξ_{gg}(r) deviates from the power law, rapidly decreases to zero, and becomes negative around 50h^{1} Mpc. However, on small scales r ≤ 1h^{1} Mpc, the solution ξ_{gg}(r) is lower than the data, even though it has already improved the Gaussian approximation (Zhang 2007). This insufficiency at r ≤ 1h^{1} Mpc is probably due to neglect of highorder nonlinear terms such as (δψ)^{3} in our perturbation. We remark that the equation of ξ_{gg}(r) has been derived assuming δψ< 1. Thus, it is only an approximation to extrapolate our calculated ξ_{gg}(r) down to smaller scales r ≤ 5h^{1} Mpc.
Fig. 1 Solution ξ_{gg}(r) compares the data of galaxies by APM (Padilla & Baugh 2003), 2dFGRS (Hawkins et al. 2003), and SDSS (Zehavi et al. 2005). Here k_{0} = 0.055h Mpc^{1} is included in the calculation. 

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We verified that the approximation of hydrostatical equilibrium can be applied in the quasilinear regime in the expanding Universe. The density fluctuation approximately behaves as δψ ∝ a(t)^{0.3}, where a(t) is the scale factor in the present stage of accelerating expansion. So the timeevolving correlation function ξ_{gg}(r,t) = ⟨ δψδψ ⟩ ∝ a^{0.6}(t) = 1 / (1 + z)^{0.6}. For the sample of ~200 000 galaxies of the SDSS (Zehavi et al. 2005), the redshift range is z = (0.02 ~ 0.167). Taking its maximum z = 0.167 into the ratio gives ξ_{gg}(r) /ξ_{gg}(r,t) ≃ (1 + 0.167)^{0.6} ~ 1.097, and the error is 0.6z ≃ 0.1. The conclusion of this analysis has also been supported by studies of numerical simulations (Hamana et al. 2001; Yoshikawa et al. 2001; Taruya et al. 2001).
Then we consider the power spectrum P(k). The power spectrum P(k) is the Fourier transform (6)of the correlation function ξ(r). It measures the massdensity fluctuation in kspace. In principle, P(k) and ξ(r) contain the same information if both are complete in their respective space, k = (0,∞), and r = (0,∞). The observed ξ_{gg}(r) is not complete, however, because it is limited to a finite range, for example, r ≤ 50 Mpc. If the observed powerlaw ξ_{gg}(r) = (r_{0}/r)^{1.8} were plugged into Eq. (6), one would have P(k) ∝ k^{1.2}, which does not comply with the observed P(k) ∝ k^{1.6} (Peacock 1999). Our P(k) is obtained from the solution ξ_{gg}(r) given on the whole range r = (0,∞). Figure 2 shows the theoretical P(k) converted by Eq. (6) from the solution ξ_{gg}(r) with the same set (a,b,c) and k_{0} as those in Fig. 1. We also show in Fig. 2 the observational data of P(k) from APM (Padilla & Baugh 2003), 2dFGRS (Cole et al. 2005), and SDSS (Blanton et al. 2004). The theoretical P(k) agrees well with the data P(k) ∝ k^{1.6} in the range of k = (0.04 ~ 0.7) h Mpc^{1}. However, at large k, the theoretical P(k) is lower than the data. This insufficiency of P(k) corresponds to that of ξ_{gg}(r) at small scales r ≤ 1h^{1} Mpc shown in Fig. 1.
Fig. 2 Power spectrum P(k) converted from ξ_{gg}(r) in Fig. 1 compares the data of APM (Padilla & Baugh 2003), 2dFGRS (Cole et al. 2005), and SDSS (Blanton et al. 2004). 

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5. Comparing the observational data of clusters
Clusters are believed to trace the cosmic mass distribution on even larger scales, and the observational data cover spatial scales that exceed those of galaxies. Now we apply the solution with the same two sets of (a,b,c) as in Sect. 4 to the system of clusters. A cluster has a mass m greater than that of a galaxy. This leads to a higher overall amplitude of ξ_{cc}(r). In addition, to match the observational data of clusters, a low value k_{0} = 0.03 Mpc^{1} is required, lower than that for galaxies.
Fig. 3 Solution ξ_{cc}(r) for clusters compares the data of SDSS clusters of two types of richness (Bahcall et al. 2003). Here (a,b,c) are the same for galaxies. But k_{0} = 0.03h Mpc^{1} is taken for clusters, smaller than that for galaxies. 

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For each set of values of (a,b,c), two solutions ξ_{cc}(r) with different amplitudes are given in Fig. 2 and are compared with two sets of data of richness N> 10 and N> 20 from the SDSS (Bahcall et al. 2003). Interpreted by Eq. (5), the N> 20 clusters have a greater m than the N> 10 clusters. The solutions match the data in the whole range r = (4 ~ 100)h^{1} Mpc.
It has long been known that there is a scaling behavior of cluster correlation. The correlation scale increases with the mean spatial separation between clusters (Szalay & Schramm 1985; Bahcall & West 1992; Bahcall 1999; Croft et al. 1997; Gonzalez et al. 2002). If a power law ξ_{cc} = (r_{0}/r)^{1.8} were used to fit the data, the “correlation length” would be of a form r_{0} ≃ 0.4d_{i}, where and n_{i} is the mean number density of clusters of type i. Simulations have also produced this r_{0} − d_{i} dependence (Bahcall & Cen 1992). For SDSS, the scaling can also be fitted by r_{0} ≃ 2.6d_{i}^{1 / 2} (Bahcall et al. 2003) and for the 2df galaxy groups by r_{0} ≃ 4.7d_{i}^{0.32} (Zandivarez et al. 2003). This kind of universal scaling of r_{0} − d_{i} has been a theoretical challenge (Bahcall 1997), and was thought to be either caused by a fractal distribution of galaxies and clusters (Szalay & Schramm 1985) or by the statistics of rare peak events (Kaiser 1984). The difference in the scaling slope was attributed to the different richness of clusters (Bahcall 1997). In our theory the scaling is fully embodied in the solution ξ_{cc}(k_{0}r) with the characteristic wave number . To comply with the empirical powerlaw, we take the theoretical “correlation length” as , where ξ_{cc} is the theoretical solution and depends on d. Figure 4 shows that the solution ξ_{cc} with k_{0} = 0.03h Mpc^{1} gives the universal scaling r_{0}(d) ≃ 0.4d, which agrees well with the observation (Bahcall 1999). If a greater k_{0} = 0.055h Mpc^{1} is taken, the solution ξ_{cc} would yield a flatter scaling r_{0}(d) ≃ 0.3d, which better fits the data of APM clusters (Bahcall et al. 2003). Our analysis based on the solution ξ_{cc} reveals that a higher background density ρ_{0} predicts a flatter slope of the scaling r_{0}(d). To conclude, the universal r_{0} − d_{i} scaling is naturally explained by the solution ξ_{cc}(r).
Fig. 4 Solution ξ_{cc}(r) with k_{0} = 0.03h Mpc^{1} gives the universal scaling r_{0} ≃ 0.4d. If a greater k_{0} = 0.055h Mpc^{1} is taken, ξ_{cc} would give a flatter scaling r_{0} ≃ 0.3d, which better fits the data of APM clusters (Bahcall et al. 2003). 

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6. Conclusions and discussions
We have presented a field theory of density fluctuations of Newtonian gravitating systems and applied it to the study of correlation functions of galaxies and clusters. We obtained the field equation Eq. (5) of the twopoint correlation function as a main result by starting from the field equation Eq. (1) of mass density, under the condition of hydrostatic equilibrium and keeping to the nonlinear order of (δψ)^{2} in perturbation, by taking the functional derivative. In deriving Eq. (5), we adopted the KirkwoodGrothPeebles ansatz necessary to cut off the hierarchy in npoint correlation functions and renormalized this to absorb divergences into the parameters. This analytic approach from first principles is different from those using gravitational potential. Equation (5) clearly explains the observational data of clustering of galaxies and clusters
on large scales. To extend the analytic study further, higherorder nonlinear terms should be included to describe smallscale clusterings better, and the cosmic evolution should be taken into consideration as a more realistic model.
Acknowledgments
Y. Zhang is supported by NSFC No. 11275187, NSFC 11421303, SRFDP, and CAS, the Strategic Priority Research Program “The Emergence of Cosmological Structures” of the Chinese Academy of Sciences, Grant No. XDB09000000.
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All Figures
Fig. 1 Solution ξ_{gg}(r) compares the data of galaxies by APM (Padilla & Baugh 2003), 2dFGRS (Hawkins et al. 2003), and SDSS (Zehavi et al. 2005). Here k_{0} = 0.055h Mpc^{1} is included in the calculation. 

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In the text 
Fig. 2 Power spectrum P(k) converted from ξ_{gg}(r) in Fig. 1 compares the data of APM (Padilla & Baugh 2003), 2dFGRS (Cole et al. 2005), and SDSS (Blanton et al. 2004). 

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In the text 
Fig. 3 Solution ξ_{cc}(r) for clusters compares the data of SDSS clusters of two types of richness (Bahcall et al. 2003). Here (a,b,c) are the same for galaxies. But k_{0} = 0.03h Mpc^{1} is taken for clusters, smaller than that for galaxies. 

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In the text 
Fig. 4 Solution ξ_{cc}(r) with k_{0} = 0.03h Mpc^{1} gives the universal scaling r_{0} ≃ 0.4d. If a greater k_{0} = 0.055h Mpc^{1} is taken, ξ_{cc} would give a flatter scaling r_{0} ≃ 0.3d, which better fits the data of APM clusters (Bahcall et al. 2003). 

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In the text 