Weak-Lensing Surface Mass Density
Weak-Lensing Surface Mass Density
This page describes the projected surface mass density and the related weak-lensing observables — convergence and tangential shear — implemented by NcWLSurfaceMassDensity. The three-dimensional mass distribution is supplied by a halo density profile NcHaloDensityProfile, and the geometry by NcDistance.
Projected Surface Mass Density
The projected surface mass density is the line-of-sight integral of the three-dimensional density profile \(\rho(r)\), \[ \Sigma(R) = \int \mathrm{d}\chi\,\rho\!\left(\sqrt{R^2 + \chi^2}\right), \] where \(R\) is the projected (two-dimensional) distance from the halo center and \(\chi\) is the distance along the line of sight, so that \(r^2 = R^2 + \chi^2\).
The mean surface mass density within a circular aperture of radius \(R\) is \[ \overline{\Sigma}(<R) = \frac{2}{R^2}\int_0^R \mathrm{d}R^\prime\,R^\prime\,\Sigma(R^\prime). \]
Convergence and Shear
The convergence \(\kappa(R)\) and tangential shear \(\gamma(R)\) are \[ \kappa(R) = \frac{\Sigma(R)}{\Sigma_\mathrm{crit}}, \qquad \gamma(R) = \frac{\Delta\Sigma(R)}{\Sigma_\mathrm{crit}} = \frac{\overline{\Sigma}(<R) - \Sigma(R)}{\Sigma_\mathrm{crit}}, \] where the critical surface density is \[ \Sigma_\mathrm{crit} = \frac{c^2}{4\pi G}\frac{D_s}{D_l D_{ls}}, \] with \(c\) the speed of light, \(G\) the gravitational constant, and \(D_s\), \(D_l\), \(D_{ls}\) the angular diameter distances to the source, to the lens, and between lens and source, respectively.
For background on cluster weak-lensing mass measurements, see Mandelbaum et al. (2006), Umetsu et al. (2012), Applegate et al. (2014), Melchior et al. (2017), and Parroni et al. (2017).
API Reference
See NcWLSurfaceMassDensity for the full class reference. The most relevant methods are:
- surface mass density:
nc_wl_surface_mass_density_sigma,nc_wl_surface_mass_density_sigma_mean - convergence and shear:
nc_wl_surface_mass_density_convergence,nc_wl_surface_mass_density_shear - critical surface density:
nc_wl_surface_mass_density_sigma_critical