The relation between the two-point and the three-point correlation functions in the non-linear gravitational clustering regime

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The connection between the two-point and the three-point correlation functions in the non-linear gravitational clustering regime is studied. Under a scaling hypothesis, we find that the three-point correlation function, ζ, obeys the scaling law ζ∝ ξ³ᵐ⁺⁴ʷ⁻²ε/²ᵐ⁺²ʷ in the nonlinear regime, where ξ, m, w, and ε are the two-point correlation function, the power index of the power spectrum in the nonlinear regime, the number of spatial dimensions, and the power index of the phase correlations, respectively. The new formula reveals the origin of the power index of the three-point correlation function. We also obtain the theoretical condition for which the ``hierarchical form'' ζ∝ξ² is reproduced.