Probabilistic estimators of Lagrangian shape biases: Universal relations and physical insights

Open Access

Table A.1

General notation employed throughout the article.

Notation	Definition
x	Eulerian positions
q	Lagrangian positions
T	Lagrangian tidal-field
K	Traceless Lagrangian tidal-field
ℐ	Shape tensor
$\tilde{I}$ $\[\tilde{\mathcal{I}}\]$	Reduced shape tensor
I	Traceless shape tensor
$\tilde{I}$ $\[\tilde{\mathbf{I}}\]$	Traceless reduced shape tensor
f	Finite-scale density bias function
F	Large-scale density bias function
p(T)	P.d.f of DM tidal field.
p_g(T)	P.d.f of DM tidal field at galaxy locations.
J₂	$δ K_{i j}^{K}$ $\[\delta K_{i j}^{K}\]$
J₂₂	$S_{22} (δ_{i j} δ_{k l}^{K})$ $\[S_{22}\left(\delta_{i j} \delta_{k l}^{K}\right)\]$
J₂₌₂	$S_{22} (δ_{i l}^{K} δ_{k j}^{K}) - \frac{1}{3} J_{22}$ $\[S_{22}\left(\delta_{i l}^{K} \delta_{k j}^{K}\right)-\frac{1}{3} \mathbf{J}_{22}\]$
b₁	Lagrangian linear bias parameter.
c_K	Linear shape-bias parameter.
c_δK, c_K⊗K	Second-order shape-bias parameters.

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