The Cusp/Core Suite




This suite is a set of recovery tests for strong lensing. We attempt to correctly recover the mass distribution in a set of spherical galaxy cluster lenses (each with a different density distribution), a triaxial galaxy from a N-body cosmological hydrodynamical simulation, and an elliptical galaxy formed from two merging spirals. The details for each are given in the table below.

Included physics

Strong gravitational lens reconstruction.


The strong lensing problem is inherently degenerate. Many different mass models will perfectly fit precisely the same data. These tests set out to determine what quality of data is required to break these degeneracies and obtain an accurate measure of the mass distribution. The most significant degeneracy is the steepness degeneracy (see references below) and it is known that this is broken by multiple sources with wide redshift separation, by time delays (if the Hubble constant is known), and/or by an independent mass measure for the lens (that can come, for example, from stellar velocity dispersions). However, there are also higher order degeneracies involving twisting of the mass contours, or varying ellipticity with radius. We examine also the influence of these degeneracies and what quality of data is required to break them.

Publications using this test

None so far. But see useful references below.

Useful references

Suggested background reading:

  • Falco et al. 1985 | first derivation of the steepness (also called the mass-sheet) degeneracy.
  • Gorenstein et al. 1988 | further discussion of major degeneracies.
  • Saha 2000 | review of lensing degeneracies, in particular the the steepness degeneracy.
  • Saha & Williams 2006 | discussion of higher-order lens degeneracies – twisting and shape degeneracies.
  • Liesenborgs et al. 2008 | generalisation of the steepness degeneracy that leads to a new ring degeneracy.

Details of setup

Units and constants

We adopt the following system of units:


Initial Conditions

The Sphere models

The Sphere models were set up with a spherical dark matter density profile given by:


where Graph and Graph are the total mass and scale length, and Graph is the central log-slope.

We set up three models: Sphere_0, Sphere_1 and Sphere_2, corresponding to a central log-slope of Graph, respectively. The total masses for each were chosen to ensure a fixed mass within 300kpc. All three were initialised with Graph particles using a monte-carlo method as in Read et al. 2006. (First, particles were drawn from the above density distribution. Then, they had their velocities initialised from a numerically calculated distribution function, assuming isotropy.) Each was also initialised with a spherical, isotropic, massless, tracer population of Graph star particles. The stars were described also by the equation above with Graph and Graph. The raw particle data was stored in the Tipsy file format.

The details for each of the models is as follows:

Model Graph Graph Graph
Sphere_0 Graph Graph Graph
Sphere_1 Graph Graph Graph
Sphere_2 Graph Graph Graph

The following figures show the initial density, surface density, cumulative mass, projected velocity dispersion (dark matter) and projected velocity dispersion (stars) for each of the models (Sphere_0 black, Sphere_1 red, Sphere_2 blue). Click on each of the images to enlarge.

To produce images, we projected each of Sphere_0,1,2 onto a 2D grid along the z-axis. We put the lens at a redshift Graph and used the following source properties (double means two images; quad means 4, but there are no quads for a spherically symmetric lens):

Label Source Images
S1 1 (Graph) double (d)
S2 2 (Graph) dd
S3 2 (Graph) dd
S4 2 (Graph) dd

We found the images for these source/lens configurations using XXX. The image positions and time delays were stored in PixeLens format.

The model data are given in the table below.

The Triaxial galaxy

This is a triaxial galaxy taken from a N-body hydrodynamical cosmological simulation (see Read et al. 2007).

The following figures show the initial density, surface density, cumulative mass, projected velocity dispersion (dark matter), and projected velocity dispersion (stars) for three different projections of the model (x black, y dotted and z dashed). Click on each of the images to enlarge.


The model data are given in the table below.

The elliptical merger remnant

J. to add here.


These files contain all of the data described above. The Tipsy files contain the raw particle data (dark matter and stars). The .tar files contain all of the ASCII data files for each model. N.B. ignore the star particle masses in the Tipsy files. The stars should be treated as massless tracers.

The .tar.gz files contain the following ASCII formatted data:

Description Format File
Projected velocity dispersion (dark matter) Graph _sigpz.txt
Projected velocity dispersion (stars) Graph _sigpzstar.txt
Projected surface density Graph _sdenz.txt
Spherically averaged density Graph _den.txt
Cumulative mass Graph _mass.txt
Image positions / mag / time delays J. to add J. to add
Source positions J. to add J. to add

How to analyse the test?

For analysis we compare input and recovered projected density distributions. We also use this as an opportunity to test the deprojection algorithm outlined in Saha, Read and Williams 2006 and refined in (coming soon). We will release a public version of this deprojection code soon.

Tests run so far

Code Person Publication
PixeLens Justin Read


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