  In the context of gravitational lensing, the density profile of lensing
galaxies is often considered to be perfectly elliptical. Potential angular
structures are generally ignored, except to explain flux ratios anomalies.
Surprisingly, the impact of azimuthal structures on extended images of the
source has not been characterized, nor its impact on the H0 inference. We
address this task by creating mock images of a point source embedded in an
extended source, lensed by an elliptical galaxy on which multipolar components
are added to emulate boxy/discy isodensity contours. Modeling such images with
a density profile free of angular structure allow us to explore the
detectability of image deformation induced by the multipoles in the residual
frame. Multipole deformations are almost always detectable for our highest
signal-to-noise mock data. However the detectability depends on the lens
ellipticity and Einstein radius, on the S/N of the data, and on the specific
lens modeling strategy. Multipoles also introduce small changes to the time
delays. We therefore quantify how undetected multipoles would impact H0
inference. When no multipoles are detected in the residuals, the impact on H0
for a given lens is in general less than a few km/s/Mpc, but in the worst case
scenario, combining low S/N in the ring and large intrinsic boxyness/discyness,
the bias on H0 can reach 10-12 km/s/Mpc. If we now look at the inference on H0
from a population of lensing galaxies, having a distribution of multipoles
representative of what is found in the light-profile of elliptical galaxies, we
then find a systematic bias on H0 < 1%. The comparison of our mock systems to
the state-of-the-art time delay lens sample studied by the H0LiCOW and TDCOSMO
collaborations, indicates that multipoles are currently unlikely to be a source
of substantial systematic bias on the inferred value of H0 from time-delay
lenses.
