Go to previous page Go up Go to next page

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 q12 =_q(x1,x2) is defined also in terms of the joint probability of the pair of objects located in the volume elements of dV1 and dV2,
dP = n2dV dV [1 + q ], (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, z123 =_ z(x1, x2,x3) and j1234 =_j(x1,x2, x3,x4), in a straightforward manner:
3 dP123 = n dV1dV2dV3 [1 + q12 + q23 + q31 + z123], (87) dP1234 = n4dV1dV2dV3dV4[1 + q12 + q13 + q14 + q23 + q24 + q34 + z123 + z124 + z134 + z234 + q12q34 + q13q24 + q14q23 + j1234], (88)
Apparently q12, z123, and j1234 are symmetric with respect to the change of the indices. Define the following quantities with the same symmetry properties:
Z =_ q q + q q + q q , (89) 123 12 23 21 13 23 31 A1234 =_ q12q23q34 + q23q34q41 + q24q41q12 + q13q32q24 + q32q24q41 + q24q41q13 + q12q24q43 + q24q43q31 + q31q12q24 + q13q34q42 + q34q42q21 + q42q21q13, (90) B1234 =_ q12q13q14 + q21q23q24 + q31q32q34 + q41q42q43, (91) C1234 =_ z123(q14 + q24 + q34) + z134(q12 + q32 + q42) + z124(q31 + q32 + q34) + z234(q12 + q13 + q14). (92)
Then it is not unreasonable to suspect that the following relations hold:
z123 = Q Z123, (93) j1234 = Ra A1234 + Rb B1234, (94) j1234 = 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 0.1 h-1 Mpc <~ r <~ 10h- 1 Mpc, } Ra = 2.5± 0.6, -1 - 1 (96) Rb = 4.3 ± 1.2 for 0.5 h Mpc <~ r <~ 4h Mpc.
The generlization of those relations for N-point correlation functions is suspected to hold generally,
sum sum N prod -1 qN (r1, ...,rN ) = QN,j q(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].
Go to previous page Go up Go to next page