The unsmoothed normalized higher order cumulants or $S_N = \langle \delta ^N \rangle _c/\langle \delta ^2 \rangle ^{N-1}_c$ can be expressed in terms of the tree-level vertices denoted as νN using the following expressions (Bernardeau et al. 2002a):
(20)$$\begin{eqnarray*}
&amp;&amp; S_3 = 3\nu _2; \quad S_4 = 4\nu _3+ 12\nu _2^2; \quad S_5 = 5\nu _4+ 60\nu _3\nu _2 + 60\nu _2^3. \quad
\end{eqnarray*}$$The vertices νN are the angular averages of the mode-coupling kernels FN defined in equation (1c) i.e. νN = N!〈 FN〉 introduced in Section   2.2 in the Fourier domain.
(21)$$\begin{eqnarray*}
&amp;&amp; \nu _N = N!\langle F_N \rangle = N! \int {\mathrm{ d}{{\bf \widehat{\Omega }}}_{k_1} \over 4\pi } \cdots \int {\mathrm{ d}{{\bf \widehat{\Omega }}}_{k_N} \over 4\pi } F_N({\bf k}_1.\cdots , {\bf k}_N);\quad \mathrm{ d}{{\bf \widehat{\Omega }}}_k = \sin \theta _k \mathrm{ d}\theta _k \mathrm{ d}\varphi _k.
\end{eqnarray*}$$The following generating function approach was introduced in Bernardeau (1992, 1994b). The generating functions ${\cal G}_{\delta }(\tau)$ are solved using the equations of gravitational dynamics encapsulated in Euler–Continuity–Poisson equations. Here, τ plays the role of a dummy variable. In the perturbative regime, the νN parameter can be computed for an arbitrary N(22)$$\begin{eqnarray*}
&amp;&amp; {\cal G}_{\delta }(\tau) = \sum _{n}{\nu _N \over N!}\tau ^N = -\tau + {12 \over 14} \tau ^2 - {29 \over 42} \tau ^3 + {79 \over 147} \tau ^4 - {2085 \over 5096} \tau ^5 + \cdots
\end{eqnarray*}$$Next, using equation (20), the one-point cumulants in 2D (Munshi et al. 1999b), denoted as ΣN as opposed to SN parameters which represent the cumulants in 3D, can be used to compute the cumulants to arbitrary order in 2D (Munshi et al. 1999b)
(23)$$\begin{eqnarray*}
&amp;&amp; \Sigma _3 = {36\over 7}; \quad \Sigma _4={2540 \over 49}; \quad \Sigma _5={793}; \quad \Sigma _6=16370;
\end{eqnarray*}$$The generalization of the one-point cumulants i.e the SN parameters to the two-point cumulant correlators $C_{pq} = \langle \delta ^p_1\delta _2^q\rangle _c/\langle \delta ^2\rangle _c^{p+q-1}\langle \delta _1\delta _2\rangle _c$ or Cpq parameters was introduced in Bernardeau (1996a). The lower order normalized cumulant correlators can also be expressed in terms of the tree-level vertices νN just as the one-point cumulants introduced in equation (20)
(24)$$\begin{eqnarray*}
&amp;&amp; C_{21} = 2\nu _2; \quad C_{31} = 3\nu _3+ 6\nu _2^2; \quad C_{41} = 4\nu _4 + 36 \nu _3\nu _2 + 24 \nu _2^3.
\end{eqnarray*}$$The corresponding quantities for convergence maps can be defined in an analogous manner and are denoted with a superscript κ i.e. $C^{\kappa }_{pq}$. To compare with observed or simulated data smoothing of the field is necessary. The smoothed generating function ${\cal G}^s_{\delta }$ can be computed from the unsmoothed generating function ${\cal G}_{\delta }$. The generating functions ${\cal G}^s_{\delta }$ and ${\cal G}_{\delta }$ are related by the following implicit relation (Bernardeau 1995):
(25)$$\begin{eqnarray*}
&amp;&amp; {\cal G}_{\delta }^s(\tau)= {\cal G}_{\delta }(\tau [1+ {\cal G}_{\delta }^s]^{-(2+n)/4}).
\end{eqnarray*}$$A top hat smoothing window is assumed and the power spectrum is approximated locally as a power law P(k) ∝ kn (Bernardeau 1995; Munshi et al. 1999b). For other window functions, e.g. Gaussian window generic results are not possible for arbitrary N. However, an order-by-order approach can be adopted to obtain the lower order cumulants (Matsubara 2003). Notice that the smoothed power law depends on the spectral index while unsmoothed vertices depend solely on the gravitational collapse in 3D spherical or cylindrical in 2D. The smoothed vertices can be recovered by Taylor-expanding the smoothed generating function ${\cal G}^s$. Using these vertices, it is possible to now compute the 2D skewness Σ3 and kurtosis Σ4 can be computed (Munshi et al. 1999b)
(26a)$$\begin{eqnarray*}
&amp;&amp; \Sigma _3 = {36 \over 7} -{3\over 2} (n+2);
\end{eqnarray*}$$(26b)$$\begin{eqnarray*}
&amp;&amp; \Sigma _4 = {2540 \over 49} - 33(n+2) + {21 \over 4}(n+2)^2.
\end{eqnarray*}$$
Citation: Munshi et al. 1999b
Citation function: Uses
Citation: Munshi et al. 1999b
Citation function: Uses; Uses
Citation: Munshi et al. 1999b
Citation function: Uses
Citation: Munshi et al. 1999b
Citation function: Uses; Uses