Probabilistic Lagrangian bias estimators and the cumulant bias expansion

Open Access

Table 1.

Orthogonal basis tensors that we consider here.

Tensor	Index Representation	Norm²	ϕ contr.
J₂	S₂(δ_ij)	3	δ
J₂₂	S₂₂(δ_ijδ_kl)	9	δ²
J_2 = 2	$S_{22} (δ_{ik} δ_{jl}) - \frac{1}{3} S_{22} (δ_{ij} δ_{kl})$ $S_{22}(\delta_{ik}\delta_{jl}) - \frac{1}{3} S_{22}(\delta_{ij}\delta_{kl})$	5	K²
J₄	S₄(δ_ijδ_kl)	5	L
J_3 − 3	S₃₃(δ_ijδ_klδ_mn)	$\frac{25}{3}$ $\frac{25}{3}$	(∇δ)²
J_3 ≡ 3	$S_{33} (δ_{ik} δ_{mj} δ_{\ln}) - \frac{3}{5} S_{33} (δ_{ij} δ_{kl} δ_{mn})$ $S_{33}(\delta_{ik}\delta_{mj}\delta_{ln}) - \frac{3}{5} S_{33}(\delta_{ij}\delta_{kl}\delta_{mn})$	7	S_3 ≡ 3
J₂₄	S₂₄(δ_ijδ_klδ_mn)	15	δL
J_2 = 4	$S_{24} (δ_{ik} δ_{jl} δ_{mn}) - \frac{1}{3} S_{24} (δ_{ij} δ_{kl} δ_{mn})$ $S_{24}(\delta_{ik}\delta_{jl}\delta_{mn}) - \frac{1}{3} S_{24}(\delta_{ij}\delta_{kl}\delta_{mn})$	$\frac{35}{6}$ $\frac{35}{6}$	ϕ_2 = 4

Notes. The first column indicates the label we use for the tensor, the second column one possible way of defining it in index notation, the third column the square of its norm (as the full contraction with itself) and the last column the term that is obtained when fully contracting it with the corresponding derivatives of the potential as in the bias expansion.

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