Operators of mathematical physics

Spin-changing operators of mathematical physics (Proposition VI.8).

Each is a spin-changing multiplier, i.e. a generalised lifted convolution \(\mathcal{C}_{s\to s'}[\Psi_T]\) with a (generally distributional) canonical kernel:

• the spin raising and lowering operators \(\eth\) and \(\bar\eth\) (Eqs. 3, 5), with \(c_\ell = \sqrt{(\ell-s)(\ell+s+1)}\) and \(-\sqrt{(\ell+s)(\ell-s+1)}\);

• the \(E\)/\(B\) decomposition of a spin \(\pm2\) polarisation field, \(a_E = -\tfrac12(a_2 + a_{-2})\) and \(a_B = \tfrac{\mathrm{i}}{2}(a_2 - a_{-2})\), each a sum of a \(\mathcal{C}_{2\to0}\) and a \(\mathcal{C}_{-2\to0}\) operator;

• the spherical Kaiser–Squires relation of weak gravitational lensing, the \(\mathcal{C}_{0\to2}\) multiplier \(\mathcal{D}_\ell = -\sqrt{(\ell+2)(\ell-1)/(\ell(\ell+1))}\) from convergence to shear.

s2conv.convolutions.physics.eth_multiplier(L: int, spin: int) → ndarray

Multiplier of the spin raising operator \(\eth\) from spin \(s\) to \(s + 1\).

Parameters:

• L (int) – Harmonic band-limit.

• spin (int) – Input spin \(s\).

Returns:

\(c_\ell = \sqrt{(\ell-s)(\ell+s+1)}\), [L].

Return type:

np.ndarray

s2conv.convolutions.physics.ethbar_multiplier(L: int, spin: int) → ndarray

Multiplier of the spin lowering operator \(\bar\eth\) from spin \(s\) to \(s - 1\).

Parameters:

• L (int) – Harmonic band-limit.

• spin (int) – Input spin \(s\).

Returns:

\(c_\ell = -\sqrt{(\ell+s)(\ell-s+1)}\), [L].

Return type:

np.ndarray

s2conv.convolutions.physics.eth(flm, spin: int) → Array

The spin raising operator \(\eth\), from spin \(s\) to \(s + 1\) (Proposition VI.8(a)).

With the conventions of Eq. (3), \(\eth = -\sin^s\theta\, (\partial_\theta + \mathrm{i}\sin^{-1}\theta\, \partial_\varphi)\, \sin^{-s}\theta\).

Parameters:

• flm – Spin-\(s\) harmonic coefficients, [..., L, 2L - 1].

• spin (int) – Input spin \(s\).

Returns:

Spin-\((s+1)\) harmonic coefficients.

Return type:

jnp.ndarray

s2conv.convolutions.physics.ethbar(flm, spin: int) → Array

The spin lowering operator \(\bar\eth\), from spin \(s\) to \(s - 1\) (Proposition VI.8(a)).

Parameters:

• flm – Spin-\(s\) harmonic coefficients, [..., L, 2L - 1].

• spin (int) – Input spin \(s\).

Returns:

Spin-\((s-1)\) harmonic coefficients.

Return type:

jnp.ndarray

s2conv.convolutions.physics.eb_decomposition(p_lm, p_conj_lm=None) → tuple

\(E\) and \(B\) components of a polarisation field (Proposition VI.8(b)).

\(a_E = -\tfrac12(a_2 + a_{-2})\) and \(a_B = \tfrac{\mathrm{i}}{2}(a_2 - a_{-2})\), each the sum of a \(\mathcal{C}_{2\to0}\) and a \(\mathcal{C}_{-2\to0}\) map.

Parameters:

• p_lm – Spin-2 coefficients \(a_2\) of \(Q + \mathrm{i}U\), [..., L, 2L - 1].

• p_conj_lm (optional) – Spin \(-2\) coefficients \(a_{-2}\) of \(Q - \mathrm{i}U\). Defaults to the conjugate of the spin-2 field, i.e. real \(Q\) and \(U\).

Returns:

Scalar harmonic coefficients of \(E\) and \(B\).

Return type:

tuple

s2conv.convolutions.physics.eb_to_polarisation(e_lm, b_lm) → tuple

Spin \(\pm2\) polarisation from \(E\) and \(B\), the inverse of eb_decomposition().

Parameters:

• e_lm – Scalar harmonic coefficients of \(E\), [..., L, 2L - 1].

• b_lm – Scalar harmonic coefficients of \(B\).

Returns:

Coefficients \(a_{\pm2} = -(a_E \pm \mathrm{i}\, a_B)\) of the spin \(\pm2\) fields, for \(\ell \ge 2\).

Return type:

tuple

s2conv.convolutions.physics.kaiser_squires_multiplier(L: int) → ndarray

Multiplier of the spherical Kaiser–Squires relation from convergence to shear.

Parameters:

L (int) – Harmonic band-limit.

Returns:

\(\mathcal{D}_\ell = -\sqrt{(\ell+2)(\ell-1)/(\ell(\ell+1))}\) for \(\ell \ge 2\), and zero below, [L].

Return type:

np.ndarray

s2conv.convolutions.physics.kaiser_squires(kappa_lm) → Array

Shear from convergence, the \(\mathcal{C}_{0\to2}\) multiplier \(\mathcal{D}_\ell\) (Proposition VI.8(c)).

Parameters:

kappa_lm – Scalar harmonic coefficients of the convergence, [..., L, 2L - 1].

Returns:

Spin-2 harmonic coefficients of the shear.

Return type:

jnp.ndarray

s2conv.convolutions.physics.kaiser_squires_inverse(gamma_lm) → Array

Convergence from shear, the \(\mathcal{C}_{2\to0}\) multiplier \(\mathcal{D}_\ell^{-1}\) (Proposition VI.8(c)).

The monopole and dipole of the convergence are not recoverable from the shear and are returned as zero.

Parameters:

gamma_lm – Spin-2 harmonic coefficients of the shear, [..., L, 2L - 1].

Returns:

Scalar harmonic coefficients of the convergence.

Return type:

jnp.ndarray