2 where v (0) k = $^{H$_{k}$}$− $_{1}$( ν ) is the Gaussian contribution to the MFs. Primordial non-Gaussianities generated by most of the infla - tionary models satisfy the hierarchical orderings which we as- sume. In the previous work [34], the first-order correctionsto the MFs in a non-Gaussian field are analytically derived: tests of the analytic formulas are given in § 5, and our results are summarized in § 6. The MFs are morphological descriptors to study statistical properties of random fields. For a smooth scalar field f ( $^{x}$) with zero mean, 〈 f 〉 = 0, the excursion set over a given thresh- old ν , Q = { x | f ( $^{x}$) > νσ } are defined, where σ ≡ 〈 f $^{2}$〉1 / 2 is the rms of the field. The MFs for a set Q in a 2-dimensional space with smooth boundary ∂ Q are given by [27] where are the skewness parameter and its derivatives. Fundamental techniques to evaluate the next-order term, v (2) k , are mostly found in [34]. We outline below the higher- order extensions of the calculation in [34]. First we define t he normalized cumulants of the field derivatives, where da and d ℓ denote the surface element of Q and the line element along ∂ Q , respectively, and κ is the geodesic cur- vature on the boundary ∂ Q . Apart from numerical factors, Minkowski functionals V$_{0}$ , V$_{1}$ , V$_{2}$ correspond to the area of Q , the length of ∂ Q , and the curvature integral of ∂ Q , respec- tively. According to the Gauss-Bonnet theorem, 2 π times the Euler characteristic χ of Q is equal to the integral of geodesic curvature of ∂ Q plus the integral of Gaussian curvature of Q . Therefore, when the excursion set is defined on the sphere of radius R , the Euler characteristic is given by a linear combina- tion of the Minkowski functionals via χ = V$_{2}$ + V$_{0}$ / 2 π R 2 [27]. In a flat space ( R → ∞ ), χ = V$_{2}$ . (which corresponds to the quantity denoted by ˆ M ( n ) µ$_{1}$ ··· µ$_{n}$ in [34]), where f$_{µ}$ denotes all the field derivatives, ( f$_{µ}$ ) = ( f , f$_{;1}$ , f$_{;2}$ , f$_{;11}$ , f$_{;22}$ , f$_{;12}$ , . . . ), and “;” indicates the covariant derivative. In the following we adopt a notation, f$_{0}$ = f , f$_{1}$ = f$_{;1}$ , f$_{2}$ = f$_{;2}$ , f$_{11}$ = f$_{;11}$ , and the corresponding index runs over µ = 0 , 1 , 2 , 11. The Eq. (9) is of zero-th order in σ for hierarchical orderings. For any statistic which is locally defined by some function F of field derivatives f$_{µ}$ , the expec- tation value of the statistic is expanded as (Eq. (22) of [34]) where σ$_{1}$ ≡ 〈 ( ∇ u ) $^{2}$〉1 / 2 is the variance of the gradient field, and ω$_{k}$ ≡ π k / $^{2}$/ Γ ( k / 2 + 1) is the volume of the unit ball in k dimensions, i.e., ω$_{0}$ = 1, ω$_{1}$ = 2, ω$_{2}$ = π . For a random Gaus- sian field, the MFs are analytically given by Tomita’s formul a [41], which corresponds to where F G ≡ 〈 F 〉 $_{G}$, F G µ$_{1}$µ$_{2}$ ··· ≡ 〈 ∂$_{µ}$$_{1}$ ∂$_{µ}$$_{2}$ · · · F 〉 $_{G}$, ∂$_{µ}$ = ∂/∂ f$_{µ}$ , and 〈· · · 〉 G denotes the expectation value for the Gaussian field which has the same two-point functions as the non-Gaussian field we consider (see [34] for detail). The first two terms on RHS of Eq. (10) correspond to the Gaussian contribution and first-order non-Gaussian corrections, respectively. For the MFs, the function F is given by [34] where Θ , δ are the step function and the delta function, re- spectively, and u$_{µ}$ = f$_{µ}$ /σ . The quantities F G µ$_{1}$µ$_{2}$ ··· are explicitly Throughout this paper, we assume hierarchical orderings of the higher-order correlators, 〈 f $^{n}$〉 c ∼ O ( σ 2 n − $^{2}$), where 〈· · · 〉 c denotes the connected part, or the cumulant. In such a case, the reduced MFs v$_{k}$ ( ν ) are expanded by the variance σ as for k = 0 in Eq. (3). are the Hermite polynomials, and we define a function where v$_{k}$ = v (0) k + v (1) k σ + v (2) k σ $^{2}$+ · · · (6) $^{H}$− $_{1}$( ν ) ≡ e ν $^{2}$/2 ∫ ∞ ν d ν e − ν $^{2}$/2 = √ π 2 erfc ( ν $_{√$_{2}$}$) , (5) H$_{n}$ ( ν ) = e ν $^{2}$/2 ( − d d ν ) n e − ν $^{2}$/ $^{2}$, (4) v$_{k}$ ( ν ) = $^{H$_{k}$}$− $_{1}$( ν ) , (3) 〈 F 〉 = F $^{G}$+ 1 6 ∑ M (3) µ$_{1}$µ$_{2}$$_{µ$_{3}$}$F G µ$_{1}$µ$_{2}$$_{µ$_{3}$}$σ + [ 1 24 ∑ M (4) µ$_{1}$µ$_{2}$µ$_{3}$$_{µ$_{4}$}$F G µ$_{1}$µ$_{2}$µ$_{3}$µ$_{4}$ + 1 72 ∑ M (3) µ$_{1}$µ$_{2}$$_{µ$_{3}$}$M (3) µ$_{4}$µ$_{5}$$_{µ$_{6}$}$F G µ$_{1}$µ$_{2}$µ$_{3}$µ$_{4}$µ$_{5}$µ$_{6}$ ] σ 2 + O ( σ $^{3}$) , (10) V$_{k}$ ( ν ) = 1 (2 π ) ( k + 1) / 2 ω$_{2}$ $^{ω$_{2}$}$− $_{k}$ω$_{k}$ ( σ$_{1}$ $_{√$_{2}$$_{σ}$}$) k e − ν $^{2}$/ $^{2}$v$_{k}$( ν ) , (2) M ( n ) µ$_{1}$ ··· µ$_{n}$ ≡ σ 2 − 2 $^{n}$〈 f$_{µ}$$_{1}$ · · · f$_{µ}$$_{n}$ 〉 c (9) S = 〈 f $^{3}$〉 σ 4 , S$_{I}$ = 〈 f $^{2}$∇$^{2}$f 〉 σ $^{2}$σ 2 1 , S$_{II}$ = 2 〈|∇ f | $^{2}$∇$^{2}$f 〉 σ 4 1 , (8) v (1) k ( ν ) = S 6 H$_{k}$$_{+}$$_{2}$ ( ν ) − kS$_{I}$ 4 H$_{k}$ ( ν ) − k ( k − 1) S$_{II}$ 4 $^{H$_{k}$}$− $_{2}$( ν ) , (7) V$_{0}$ ( ν ) = ∫$_{Q}$ da , V$_{1}$ ( ν ) = 1 4 ∫$_{∂}$$_{Q}$ d ℓ, V$_{2}$ ( ν ) = 1 2 π ∫$_{∂}$$_{Q}$ κ d ℓ, (1) II. ANALYTIC MINKOWSKI FUNCTIONALS WITH SECOND-ORDER NON-GAUSSIANITY