Lensing

When two galaxies are aligned perfectly down the line-of-sight to Earth, the background galaxy’s light is bent by the intervening mass of the foreground galaxy. Its light can be fully bent around the foreground galaxy, traversing multiple paths to the Earth, meaning that the background galaxy is observed multiple times. This by-chance alignment of two galaxies is called a strong gravitational lens and a two-dimensional scheme of such a system is pictured below.

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(Image Credit: Image credit: F. Courbin, S. G. Djorgovski, G. Meylan, et al., Caltech / EPFL / WMKO, https://www.astro.caltech.edu/~george/qsolens/)

PyAutoLens is software for analysing strong lenses!

To use PyAutoLens we first import autolens and the plot module.

import autolens as al
import autolens.plot as aplt

Grids

To describe the deflection of light due to the lens galaxy’s mass, PyAutoLens uses Grid2D data structures, which are two-dimensional Cartesian grids of (y,x) coordinates.

Below, we create and plot a uniform Cartesian Grid2D (the pixel_scales describes the conversion from pixel units to arc-seconds):

grid = al.Grid2D.uniform(
    shape_native=(50, 50), pixel_scales=0.05
)
grid_plotter = aplt.Grid2DPlotter(grid=grid)
grid_plotter.figure_2d()

This is what our Grid2D looks like:

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Light Profiles

We will ray-trace this Grid2D’s (y,x) coordinates to calculate how a lens galaxy’s mass deflects the source galaxy’s light.

This requires analytic functions representing the light and mass distributions of galaxies. PyAutoLens uses Profile objects for this, such as the elliptical sersic LightProfile:

sersic_light_profile = al.lp.EllSersic(
    centre=(0.0, 0.0),
    elliptical_comps=(0.1, 0.1),
    intensity=0.05,
    effective_radius=2.0,
    sersic_index=4.0,
)

By passing this Profile a Grid2D, we can evaluate the light at every coordinate on that Grid2D, creating an image of the LightProfile:

image = sersic_light_profile.image_2d_from_grid(grid=grid)

The PyAutoLens plot module provides methods for plotting objects and their properties, like the LightProfile’s image.

light_profile_plotter = aplt.LightProfilePlotter(
    light_profile=sersic_light_profile, grid=grid
)
light_profile_plotter.figures_2d(image=True)

The light profile’s image appears as shown below:

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Mass Profiles

PyAutoLens uses MassProfile objects to represent a galaxy’s mass distribution and perform ray-tracing calculations.

Below we create an elliptical isothermal MassProfile and calculate and display its convergence, gravitational potential and deflection angles using the Cartesian grid:

isothermal_mass_profile = al.mp.EllIsothermal(
    centre=(0.0, 0.0),
    elliptical_comps=(0.1, 0.1),
    einstein_radius=1.6,
)

convergence = isothermal_mass_profile.convergence_2d_from_grid(grid=grid)
potential = isothermal_mass_profile.potential_2d_from_grid(grid=grid)
deflections = isothermal_mass_profile.deflections_2d_from_grid(grid=grid)

mass_profile_plotter = aplt.MassProfilePlotter(
    mass_profile=isothermal_mass_profile, grid=grid
)
mass_profile_plotter.figures_2d(
    convergence=True, potential=True, deflections_y=True, deflections_x=True
)

Here is how the convergence, potential and deflection angles appear:

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For anyone not familiar with gravitational lensing, don’t worry about what the convergence and potential are for now. The key thing to note is that the deflection angles describe how a given mass distribution deflects light-rays as they travel towards the Earth through the Universe.

Galaxies

A Galaxy object is a collection of LightProfile and MassProfile objects at a given redshift. The code below creates two galaxies representing the lens and source galaxies shown in the strong lensing diagram above.

lens_galaxy = al.Galaxy(
    redshift=0.5, light=sersic_light_profile, mass=isothermal_mass_profile
)

source_galaxy = al.Galaxy(redshift=1.0, light=another_light_profile)

The geometry of the strong lens system depends on the cosmological distances between the Earth, the lens galaxy and the source galaxy. It there depends on the redshifts of the Galaxy objects.

By passing these Galaxy objects to a Tracer, PyAutoLens uses these galaxy redshifts and a cosmological model to create the appropriate strong lens system.

tracer = al.Tracer.from_galaxies(
    galaxies=[lens_galaxy, source_galaxy], cosmology=cosmo.Planck15
)

Ray Tracing

We can now create the image of a strong lens system!

When calculating this image, the Tracer performs all ray-tracing for the strong lens system. This includes using the lens galaxy’s total mass distribution to deflect the light-rays that are traced to the source galaxy. As a result, the source appears as a multiply imaged and strongly lensed Einstein ring.

image = tracer.image_2d_from_grid(grid=grid)

tracer_plotter = aplt.TracerPlotter(tracer=tracer, grid=grid)
tracer_plotter.figures_2d(image=True)

This makes the image below, where the source’s light appears as a multiply imaged and strongly lensed Einstein ring.

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Extending Objects

The PyAutoLens API has been designed such that all of the objects introduced above are extensible. Galaxy objects can take many Profile’s and Tracer objects many Galaxy’s.

If the galaxies are at different redshifts a strong lensing system with multiple lens planes will be created, performing complex multi-plane ray-tracing calculations.

To finish, lets create a Tracer with 3 galaxies at 3 different redshifts, forming a system with two distinct Einstein rings! The mass distribution of the first galaxy also has separate components for its stellar mass and dark matter, where the stellar mass using a LightMassProfile representing both its light and mass.

lens_galaxy_0 = al.Galaxy(
    redshift=0.5,
    bulge=al.lmp.EllSersic(
        centre=(0.0, 0.0),
        axis_ratio=0.9,
        angle=45.0,
        intensity=0.5,
        effective_radius=0.3,
        sersic_index=2.5,
        mass_to_light_ratio=0.3,
    ),
    disk=al.lmp.EllExponential(
        centre=(0.0, 0.0),
        axis_ratio=0.6,
        angle=45.0,
        intensity=1.0,
        effective_radius=2.0,
        mass_to_light_ratio=0.2,
    ),
    dark=al.mp.SphNFW(centre=(0.0, 0.0), kappa_s=0.08, scale_radius=30.0),
)

lens_galaxy_1 = al.Galaxy(
    redshift=1.0,
    light=al.lp.EllExponential(
        centre=(0.1, 0.1), , elliptical_comps=(0.1, 0.1), intensity=3.0, effective_radius=0.1
    ),
    mass=al.mp.EllIsothermal(
        centre=(0.1, 0.1), , elliptical_comps=(0.1, 0.1), einstein_radius=0.4
    ),
)

source_galaxy = al.Galaxy(
    redshift=2.0,
    light=al.lp.EllSersic(
        centre=(0.2, 0.2),
        e1=-0.055555,
        e2=0.096225,
        intensity=2.0,
        effective_radius=0.1,
        sersic_index=1.5,
    ),
)

tracer = al.Tracer.from_galaxies(galaxies=[lens_galaxy_0, lens_galaxy_1, source_galaxy])

tracer_plotter = aplt.TracerPlotter(tracer=tracer, grid=grid)
tracer_plotter.figures_2d(image=True)

This is what the lens looks like:

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Wrap Up

If you are unfamiliar with strong lensing and not clear what the above quantities or plots mean, fear not, in chapter 1 of the HowToLens lecture series we’ll take you through strong lensing theory in detail, whilst teaching you how to use PyAutoLens at the same time! Checkout the tutorials section of the readthedocs!