content_image/441481.jpg content_image/441477.jpg content_image/441473.jpg content_image/441475.jpg content_image/441474.jpg content_image/441472.jpg content_image/441480.jpg content_image/441479.jpg content_image/441476.jpg content_image/441478.jpg
tests of the analytic formulas are given in § 5, and our results are summarized in § 6. 

## II. ANALYTIC MINKOWSKI FUNCTIONALS WITH SECOND-ORDER NON-GAUSSIANITY 

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] 

<img src='content_image/441481.jpg'>

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$_{2}$ . The expectation values of MFs in 2D are generically given 

<img src='content_image/441477.jpg'>

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 

are the Hermite polynomials, and we define a function 

for k = 0 in Eq. (3). 

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 

<img src='content_image/441473.jpg'>

<img src='content_image/441475.jpg'>

<img src='content_image/441474.jpg'>

<img src='content_image/441472.jpg'>

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: 

<img src='content_image/441480.jpg'>

where 

<img src='content_image/441479.jpg'>

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, 

(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]) 

<img src='content_image/441476.jpg'>

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 

<img src='content_image/441478.jpg'>

