Higher-order correlation functions

3.3 Higher-order correlation functions

One of the most direct methods to evaluate the deviation from Gaussianity is to compute the higher-order correlation functions. Suppose that $x i$ now labels the position of the $i$ -th object (galaxy). Then the two-point correlation function $ξ12 ≡$ $ξ (x1, x2)$ is defined also in terms of the joint probability of the pair of objects located in the volume elements of $δV1$ and $δV2$ ,

where $¯n$ is the mean number density of the objects. This definition is generalized to three- and four-point correlation functions, $ζ123 ≡ ζ(x1, x2,x3)$ and $η1234 ≡$ $η (x1, x2,x3, x4)$ , in a straightforward manner:

3 δP123 = ¯n δV1δV2δV3 [1 + ξ12 + ξ23 + ξ31 + ζ123], (87 ) δP1234 = ¯n4 δV1δV2δV3δV4 [1 + ξ12 + ξ13 + ξ14 + ξ23 + ξ24 + ξ34 + ζ123 + ζ124 + ζ134 + ζ234 + ξ12ξ34 + ξ13ξ24 + ξ14ξ23 + η1234], (88 )

Apparently $ξ12$ , $ζ123$ , and $η1234$ are symmetric with respect to the change of the indices. Define the following quantities with the same symmetry properties:

Z ≡ ξ ξ + ξ ξ + ξ ξ , (89 ) 123 12 23 21 13 23 31 A1234 ≡ ξ12ξ23ξ34 + ξ23ξ34ξ41 + ξ24ξ41ξ12 + ξ13ξ32ξ24 + ξ32ξ24ξ41 + ξ24ξ41ξ13 + ξ12ξ24ξ43 + ξ24ξ43ξ31 + ξ31ξ12ξ24 + ξ13ξ34ξ42 + ξ34ξ42ξ21 + ξ42ξ21ξ13, (90 ) B1234 ≡ ξ12ξ13ξ14 + ξ21ξ23ξ24 + ξ31ξ32ξ34 + ξ41ξ42ξ43, (91 ) C1234 ≡ ζ123(ξ14 + ξ24 + ξ34) + ζ134(ξ12 + ξ32 + ξ42) + ζ124(ξ31 + ξ32 + ξ34) + ζ234(ξ12 + ξ13 + ξ14). (92 )

Then it is not unreasonable to suspect that the following relations hold:

ζ123 = QZ123, (93 ) η1234 = RaA1234 + RbB1234, (94 ) η1234 = RcC1234, (95 )

where $Q$ , $Ra$ , $Rb$ , and $Rc$ are constants. In fact, the analysis of the two-dimensional galaxy catalogues [68 ] revealed

Q = 1.29 ± 0.21 for 0.1h−1 Mpc ≲ r ≲ 10h−1 Mpc, } Ra = 2.5 ± 0.6, −1 −1 (96 ) Rb = 4.3 ± 1.2 for 0.5h Mpc ≲ r ≲ 4h Mpc.

The generlization of those relations for $N$ -point correlation functions is suspected to hold generally,

and is called the hierarchical clustering ansatz. Cosmological $N$ -body simulations approximately support the validity of the above ansatz, but also detect the finite deviation from it [82