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,
δP = n¯2 δV δV [1 + ξ ], (86 ) 12 1 2 12
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,
∑ ∑ N∏−1 ξN (r1, ...,rN ) = QN,j ξ (rab), (97 ) j (ab)
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 [82Jump To The Next Citation Point].
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