Weak gravitational lensing, the distortion of distant galaxy images by
foreground matter, is a key tool for probing the dark sector of the Universe.
The density profiles of galaxy clusters reveal properties of dark matter such
as its self-interaction rate. The Euclid satellite will map hundreds of
thousands of massive clusters with weak lensing. The regions just outside
cluster cores, ideal for probing dark matter, are often excluded because of
higher-order distortions known as "flexion". Used as a signal, flexion can
improve cluster mass constraints by a factor of two. This PhD aims to measure
flexion for the first time on a large sample of Euclid clusters and combine it
with shear to constrain inner mass profiles and the dark matter
self-interaction rate.