Lensing (glass.lensing)

The glass.lensing module provides functionality for simulating gravitational lensing by the matter distribution in the universe.

Iterative lensing

class glass.lensing.MultiPlaneConvergence(cosmo)

Compute convergence fields iteratively from multiple matter planes.

Attributes:

delta

The current matter plane.

kappa

The current convergence plane.

wlens

The weight of the current matter plane.

zsrc

The redshift of the current convergence plane.

Methods

add_plane(delta, zsrc[, wlens])	Add a mass plane at redshift zsrc to the convergence.
add_window(delta, w)	Add a mass plane from a window function to the convergence.

glass.lensing.multi_plane_matrix(shells, cosmo)

Compute the matrix of lensing contributions from each shell.

glass.lensing.multi_plane_weights(weights, shells, cosmo)

Compute effective weights for multi-plane convergence.

Converts an array weights of relative weights for each shell into the equivalent array of relative lensing weights.

This is the discretised version of the integral that turns a redshift distribution \(n(z)\) into the lensing efficiency sometimes denoted \(g(z)\) or \(q(z)\).

Parameters:

weightsarray_like

Relative weight of each shell. The first axis must broadcast against the number of shells, and is normalised internally.

shellslist of RadialWindow

Window functions of the shells.

cosmoCosmology

Cosmology instance.

Returns:

lensing_weightsarray_like

Relative lensing weight of each shell.

Lensing fields

glass.lensing.from_convergence(kappa, lmax=None, *, potential=False, deflection=False, shear=False, discretized=True)

Compute other weak lensing maps from the convergence.

Takes a weak lensing convergence map and returns one or more of deflection potential, deflection, and shear maps. The maps are computed via spherical harmonic transforms.

Parameters:

kappaarray_like

HEALPix map of the convergence field.

lmaxint, optional

Maximum angular mode number to use in the transform.

potential, deflection, shearbool, optional

Which lensing maps to return.

Returns:

psiarray_like

Map of the deflection potential. Only returned if potential is true.

alphaarray_like

Map of the deflection (complex). Only returned if deflection if true.

gammaarray_like

Map of the shear (complex). Only returned if shear is true.

Notes

The weak lensing fields are computed from the convergence or deflection potential in the following way. [1]

Define the spin-raising and spin-lowering operators of the spin-weighted spherical harmonics as

\[\eth {}_sY_{lm} = +\sqrt{(l-s)(l+s+1)} \, {}_{s+1}Y_{lm} \;, \\ \bar{\eth} {}_sY_{lm} = -\sqrt{(l+s)(l-s+1)} \, {}_{s-1}Y_{lm} \;.\]

The convergence field \(\kappa\) is related to the deflection potential field \(\psi\) by the Poisson equation,

\[2 \kappa = \eth\bar{\eth} \, \psi = \bar{\eth}\eth \, \psi \;.\]

The convergence modes \(\kappa_{lm}\) are hence related to the deflection potential modes \(\psi_{lm}\) as

\[2 \kappa_{lm} = -l \, (l+1) \, \psi_{lm} \;.\]

The deflection \(\alpha\) is the gradient of the deflection potential \(\psi\). On the sphere, this is

\[\alpha = \eth \, \psi \;.\]

The deflection field has spin weight \(1\) in the HEALPix convention, in order for points to be deflected towards regions of positive convergence. The modes \(\alpha_{lm}\) of the deflection field are hence

\[\alpha_{lm} = \sqrt{l \, (l+1)} \, \psi_{lm} \;.\]

The shear field \(\gamma\) is related to the deflection potential \(\psi\) and deflection \(\alpha\) as

\[2 \gamma = \eth\eth \, \psi = \eth \, \alpha \;,\]

and thus has spin weight \(2\). The shear modes \(\gamma_{lm}\) are related to the deflection potential modes as

\[2 \gamma_{lm} = \sqrt{(l+2) \, (l+1) \, l \, (l-1)} \, \psi_{lm} \;.\]

References

[1]

Tessore N., et al., OJAp, 6, 11 (2023). doi:10.21105/astro.2302.01942

glass.lensing.shear_from_convergence(kappa, lmax=None, *, discretized=True)

Weak lensing shear from convergence.

Deprecated since version 2023.6: Use the more general from_convergence() function instead.

Computes the shear from the convergence using a spherical harmonic transform.

Applying lensing

glass.lensing.deflect(lon, lat, alpha)

Apply deflections to positions.

Takes an array of deflection values and applies them to the given positions.

Parameters:

lon, latarray_like

Longitudes and latitudes to be deflected.

alphaarray_like

Deflection values. Must be complex-valued or have a leading axis of size 2 for the real and imaginary component.

Returns:

lon, latarray_like

Longitudes and latitudes after deflection.

Notes

Deflections on the sphere are defined as follows: The complex deflection \(\alpha\) transports a point on the sphere an angular distance \(|\alpha|\) along the geodesic with bearing \(\arg\alpha\) in the original point.

In the language of differential geometry, this function is the exponential map.