Measuring line-of-sight shear with Einstein rings

Today I released a paper written in collaboration with Pierre Fleury, Julien Larena and Matteo Martinelli on how we can measure line-of-sight shear from Einstein rings. But what is line-of-sight shear, and why are we interested in measuring it in the first place?

To answer these questions, let’s review the phenomenon of strong gravitational lensing. Strong lensing occurs due to the local curvature of spacetime induced by massive objects. In simple terms, mass bends the spacetime around it, and light, wanting to take the quickest path anywhere (the geodesic), follows the curvature of that spacetime. The result is that we can sometimes see “behind” massive objects. If one galaxy is lined up in front of another more distant galaxy, and they are both lined up with us, the observers on Earth, we can see multiple images of the background galaxy, as its light takes different paths around the lens galaxy. If the alignment is good enough, the image can take the form of an Einstein ring, pictured below.

The Einstein ring known as the Cosmic Horseshoe, imaged by the Hubble Space Telescope. The red light in the middle is the lens galaxy while the blue ring of light comes from the more distant background galaxy.

However, as the light propagates through the Universe, it encounters other massive objects, which can also very slightly change its path, even if they don’t strongly lens the light. The result of these so-called “line-of-sight (LOS) effects” is that Einstein rings can be distorted and no longer look like perfect circles. One of the LOS effects is called convergence, which acts to rescale the size of an image. It’s not measurable, as the convergence can always be absorbed in a redefinition of the source position, a quantity which is almost never accessible from strong lensing data. This is known as the mass-sheet degeneracy.

A more interesting effect is called shear, which acts to change circles to ellipses. A novel notion of shear was found using a new formalism by Pierre, Julien and Jean-Philippe Uzan in their 2021 paper “Line-of-sight effects in strong gravitational lensing“. They proposed that this “LOS shear” would not be degenerate with the lens and source model, unlike the vast majority of other physical quantities in strong lensing.

How convergence κ and shear γ act on an image. Plot taken from Wikipedia.

The goal of my paper with Pierre, Julien and Matteo was therefore to test this conclusion. Is the LOS shear actually a measurable quantity? Or is it degenerate with the lens model?

We’re interested in answering this question because the LOS shear has the potential to become a new cosmological probe, as it encapsulates what is in effect the weak lensing of strong lensing. Weak lensing gives us information about large-scale structures in the Universe, whereas strong lensing is a smaller, galaxy-scale phenomenon. Uniting these two scales, perhaps by cross-correlation the LOS shear with weak lensing shear measurements, could reveal a great deal of information about the distribution of dark matter in the Universe.

But first we need to know if we can actually measure it.

To do this, we used lenstronomy, a public strong lensing code, to simulate mock strong lensing images. We came up with some very complicated lens and source models in order to make the images as physically realistic as possible. We simulated the noise as though we were observing these images using the Hubble Space Telescope. Our resulting mock catalogue of sixty four images is pictured below.

Our catalogue of simulated strong lensing images. Some are Einstein crosses, with four distinct images, others are horseshoes like the Cosmic Horseshoe, and others are complete rings.

We then used a Markov chain Monte Carlo (MCMC) parameter inference to fit these images with different models. We were interested to see what happens if you remove certain parameters from the models used to fit the images. We found that if you don’t account for certain aspects of the lens model, such as the alignment of the dark matter with the baryons, or if you use a very simplistic model to fit a complex image, the LOS shear is not measured very well. However, there is no systematic bias that would be evidence of a degeneracy. When fitting the images with good models that match those used to create the mocks, the LOS shear is measured very well indeed.

This is depicted in the plots below, which show the output LOS shear values found by the MCMC from the images as a function of the values used to create the image; orange and pink correspond to the better models where good fits are expected. Green and blue correspond to the deficient models where bad fits are expected. The colour bars represent the quality of the image, with darker colours corresponding to better quality images (those closest to complete rings). We see that better quality images tend to yield a more precise recovery of the shear.

Input vs output LOS shear for all the models tested (see the paper for full details).

Our paper contains many more details about the creation of the images, the models used to fit them, and plenty of statistical analysis of our results. We also checked the validity of the tidal approximation, which is one of the starting assumptions of the LOS formalism. We actually found that the approximation may not always be robust, but violating it does not seem to affect the accuracy of the measurement of the LOS shear (the precision is worsened).

This work depended a lot on the use of open-source software. We are very proud to have modified lenstronomy to include the new LOS formalism, and that our pull request for the modification was accepted a few weeks ago. The code for creating and analysing the mock images shown above is here: https://github.com/nataliehogg/analosis and the rest of the code for this work is in Jupyter notebooks here: https://github.com/nataliehogg/los_proof_of_concept.

In conclusion, we can’t wait to get out there and measure LOS shear in real data! We’re also keen to explore the higher-order LOS effects (namely flexion) and demonstrate how LOS shear can be used in cross-correlation with other probes to boost measurements of cosmological parameters.

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