Ellipticities

Definitions

epsilon-ellipticity

\[\epsilon = \frac{1 - q}{1 + q} \, e^{2i\phi}\]

chi-ellipticity

\[\chi = \frac{1 - q^2}{1 + q^2} \, e^{2i\phi}\]

Converting between definitions

cosmicshear.chi_from_epsilon(eps)

Transform epsilon-ellipticity to chi-ellipticity.

\[\chi = \frac{2\epsilon}{1 + |\epsilon|^2}\]

cosmicshear.epsilon_from_chi(chi)

Transform chi-ellipticity to epsilon-ellipticity.

\[\epsilon = \frac{\chi}{1 + \sqrt{1 - |\chi|^2}}\]

Coordinate transformations

cosmicshear.transform(jac, eps=None)

Transform epsilon-ellipticity between coordinate systems.

Uses the Jacobian matrix jac (or a stack of matrices). Transforms the ellipticity eps if given, or zero ellipticity otherwise.

Internally, the code computes the shear transformation via the decomposition

\[\mathrm{J} \begin{pmatrix} 1 + \chi_1 & \chi_2 \\ \chi_2 & 1 - \chi_1 \end{pmatrix} \mathrm{J}^{\mathsf{T}} \\ = \frac{1}{1 + |\epsilon|^2} \left[ \mathrm{J} \begin{pmatrix} 1 + \epsilon_1 & \epsilon_2 \\ \epsilon_2 & 1 - \epsilon_1 \end{pmatrix} \right] \left[ \mathrm{J} \begin{pmatrix} 1 + \epsilon_1 & \epsilon_2 \\ \epsilon_2 & 1 - \epsilon_1 \end{pmatrix} \right]^{\mathsf{T}} \;.\]

cosmicshear.inverse_transform(jac, eps=None)

Inverse-transform epsilon-ellipticity between coordinate systems.

Uses the Jacobian matrix jac (or a stack of matrices). Transforms the ellipticity eps if given, or zero ellipticity otherwise.

Equivalent to transform(inv(jac), eps), but does not compute the inverse explicitly.

Internally, the code computes the shear transformation via the decomposition

\[\mathrm{J}^{-1} \begin{pmatrix} 1 + \chi_1 & \chi_2 \\ \chi_2 & 1 - \chi_1 \end{pmatrix} \mathrm{J}^{-\mathsf{T}} \\ = \frac{(1 - |\epsilon|^2)^2}{1 + |\epsilon|^2} \left[ \begin{pmatrix} 1 - \epsilon_1 & -\epsilon_2 \\ -\epsilon_2 & 1 + \epsilon_1 \end{pmatrix} \mathrm{J} \right]^{-1} \left[ \begin{pmatrix} 1 - \epsilon_1 & -\epsilon_2 \\ -\epsilon_2 & 1 + \epsilon_1 \end{pmatrix} \mathrm{J} \right]^{-\mathsf{T}} \;.\]