Cluster-Scale Lenses

Galaxy clusters are the beasts of strong lensing. They contain tens or hundreds of lens galaxies and lensed sources, with lensed sources at many different redshifts requiring full multi-plane ray-tracing calculations. They contain one or more brightest cluster galaxy(s) a large scale dark matter halo and have arcs with Einstein Radii 10.0” -> 100.0” and beyond.

Here is an image of SDSS1152P3312, the example cluster we will adopt for illustrating PyAutoLens’s cluster modeling tools:

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Point Source

Just like for group-scale lenses, we will fit the cluster using a point-source dataset.

point_dict = al.PointDict.from_json(
    file_path=path.join(dataset_path, "point_dict.json")
)

Source-Plane Chi Squared

To model a cluster, we assume that every source galaxy is a PointSrcChi model, which means the goodness-of-fit is evaluated in the source-plane. This removes the need to iteratively solve the lens equation. However, we still define a PointSolver, incase we wish to perform image-plane fits.

point_solver = al.PointSolver(grid=grid, pixel_scale_precision=0.025)

Lens Model

A cluster scale strong lens model is typically composed of the following:

  • One or more brightest cluster galaxies (BCG), which are sufficiently large that we model them individually.
  • One or more cluster-scale dark matter halos, which are again modeled individually.
  • Tens or hundreds of galaxy cluster member galaxies. The low individual masses of these objects means we cannot
model them individually are constrain their mass, but their collectively large enough mass to need modeling. These are modeled using a scaling relation which assumes that light traces mass, where the luminosity of each individual galaxy is used to set up this scaling relation.
  • Tens or hundreds of source galaxies, each with multiple sets of images that constrain the lens model. These are

modeled as a point-source, although PyAutoLens includes tools for modeling the imaging data of sources once a good lens model is inferred. The source redshifts are also used to account for multi-plane ray-tracing.

Therefore, we again load the model from a .json file:

model_path = path.join("scripts", "group", "models")
model_file = path.join(model_path, "lens_x3__source_x1.json")

lenses_file = path.join(model_path, "lenses.json")
lenses = af.Collection.from_json(file=lenses_file)

sources_file = path.join(model_path, "sources.json")
sources = af.Collection.from_json(file=sources_file)

galaxies = lenses + sources

model = af.Collection(galaxies=galaxies)

SExtractor Catalogues

Composing the lens model for cluster scale objects requires care, given there are could be hundreds of lenses and sources galaxies. Manually writing the model in a Python script, in the way we do for galaxy-scale lenses, is therefore not feasible.

For this cluster, we therefore composed the the model by interfacing with Source Extractor (https://sextractor.readthedocs.io/) catalogue files. A full illustration of how to make the lens and source models from catalogue files is given at the following links:

These files can be easily altered to compose a cluster model suited to your lens dataset!

Lens Modeling

We are now able to model this dataset as a point source:

search = af.DynestyStatic(name="overview_clusters")

analysis = al.AnalysisPoint(point_dict=point_dict, solver=point_solver)

result = search.fit(model=model, analysis=analysis)

Result

The result contains information on the BCG, cluster scale dark matter halo and mass-light scaling relation:

print(result.max_log_likelihood_instance.galaxies.bcg.mass)
print(result.max_log_likelihood_instance.galaxies.dark.mass)
print(result.max_log_likelihood_instance.galaxies.scaling_relation)

Extended Source Fitting

For clsuter-scale lenses fitting the extended surface-brightness is extremely difficult. The models become high dimensional and difficult to fit, and it becomes very computationally. Furthermore, the complexity of cluster mass models can make it challenging to compose a mass model which is sufficiently accurate that a source reconstruction is even feasible!

Nevertheless, we are currently developing tools that try and make this possible. These will take approaches like fitting individual sources after modeling the entire cluster as a point-source and parallelizing the model-fitting process out in a way that ‘breaks-up’ the model-fitting procedure.

These tools are in-development, but we are keen to have users with real sciences cases trial them as we develop them. If you are interested please contact me! (https://github.com/Jammy2211).

Wrap-Up

The cluster package of the autolens_workspace contains numerous example scripts for performing cluster-sale modeling and simulating cluster-scale strong lens datasets.