Probabilistic Lagrangian bias estimators and the cumulant bias expansion

Open Access

Table B.1.

Covariances of various derivatives of the potential.

Example Terms	n₁	n₂	n₃	Value
⟨ϕ₁₁²⟩	4	0	0	$\frac{σ_{0}^{2}}{5}$ $\frac{\sigma_{0}^{2}}{5}$
⟨ϕ₁₁ϕ₂₂⟩, ⟨ϕ₁₂²⟩	2	2	0	$\frac{σ_{0}^{2}}{15}$ $\frac{\sigma_{0}^{2}}{15}$

⟨ϕ₁₁₁₁ϕ₁₁⟩, ⟨ϕ₁₁₁²⟩	6	0	0	$\frac{σ_{1}^{2}}{7}$ $\frac{\sigma_{1}^{2}}{7}$
⟨ϕ₁₁₁₁ϕ₂₂⟩, ⟨ϕ₁₁₁₂ϕ₁₂⟩, ⟨ϕ₁₁₂²⟩	4	2	0	$\frac{σ_{1}^{2}}{35}$ $\frac{\sigma_{1}^{2}}{35}$
⟨ϕ₁₁₂₃ϕ₂₃⟩, ⟨ϕ₁₁₂₂ϕ₃₃⟩, ⟨ϕ₁₂₃²⟩	2	2	2	$\frac{σ_{1}^{2}}{105}$ $\frac{\sigma_{1}^{2}}{105}$

⟨ϕ₁₁₁₁₁ϕ₁₁₁⟩, ⟨ϕ₁₁₁₁²⟩	8	0	0	$\frac{σ_{2}^{2}}{9}$ $\frac{\sigma_{2}^{2}}{9}$
⟨ϕ₁₁₁₁₁ϕ₁₂₂⟩, ⟨ϕ₁₁₁₂²⟩	6	2	0	$\frac{σ_{2}^{2}}{63}$ $\frac{\sigma_{2}^{2}}{63}$
⟨ϕ₁₁₁₁₂ϕ₂₂₂⟩, ⟨ϕ₁₁₁₁ϕ₂₂₂₂⟩, ⟨ϕ₁₁₂₂²⟩	4	4	0	$\frac{σ_{2}^{2}}{105}$ $\frac{\sigma_{2}^{2}}{105}$
⟨ϕ₁₁₁₁₂ϕ₂₃₃⟩, ⟨ϕ₁₁₁₂ϕ₁₂₃₃⟩, ⟨ϕ₁₁₂₃²⟩	4	2	2	$\frac{σ_{2}^{2}}{315}$ $\frac{\sigma_{2}^{2}}{315}$

Notes. The first column is an (incomplete) list of examples to give an idea of what kind of terms fall under the same combination of n₁, n₂ and n₃. All uneven terms are omitted, because they are zero and terms containing 0th and first derivatives are omitted, because the are not Galilei-invariant.

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