Definitions

The GLASS code uses the following mathematical definitions.

deflection

The deflection \(\alpha\) is a complex value with spin weight \(1\). It describes the displacement of a position along a geodesic (i.e. great circle). The angular distance of the displacement is the absolute value \(|\alpha|\). The direction of the displacement is the angle given by the complex argument \(\arg\alpha\), such that \(\arg\alpha = 0^\circ\) is north, \(\arg\alpha = 90^\circ\) is east, \(\arg\alpha = 180^\circ\) is south, and \(\arg\alpha = -90^\circ\) is west.

ellipticity

If \(q = b/a\) is the axis ratio of an elliptical isophote with semi-major axis \(a\) and semi-minor axis \(b\), and \(\phi\) is the orientation of the elliptical isophote, the complex-valued ellipticity is

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

pixel window function

The convolution kernel that describes the shape of pixels in a spherical map. No discretisation of the sphere has pixels of exactly the same shape, and the pixel window function is therefore an approximation: It is an effective kernel \(w_l\) such that the discretised map \(F\) of a spherical function \(f\) has spherical harmonic expansion

\[F_{lm} \approx w_l \, f_{lm} \;.\]

radial window

A radial window consist of a window function that assigns a weight \(W(z)\) to each redshift \(z\) along the line of sight. Each radial window has an associated effective redshift \(z_{\rm eff}\) which could be e.g. the central or mean redshift of the weight function.

A set of window functions \(W_1, W_2, \ldots\), defines the shells of the simulation.

spherical function

A spherical function \(f\) is a function that is defined on the sphere. Function values are usually parametrised in terms of spherical coordinates, \(f(\theta, \phi)\), or using a unit vector, \(f(\hat{n})\).

spherical harmonic expansion

A scalar spherical function \(f\) can be expanded into the spherical harmonics \(Y_{lm}\),

\[f(\hat{n}) = \sum_{lm} f_{lm} \, Y_{lm}(\hat{n}) \;.\]

If \(f\) is not scalar but has non-zero spin weight \(s\), it can be expanded into the spin-weighted spherical harmonics \({}_sY_{lm}\) instead,

\[f(\hat{n}) = \sum_{lm} f_{lm} \, {}_sY_{lm}(\hat{n}) \;.\]

visibility map

A visibility map describes the a priori probability of observing an object inside a given pixel, with pixel values between 0 and 1.