Image formation near hyperbolic umbilic in strong gravitational lensing

Hyperbolic umbilic (HU) is a point singularity of the gravitational lens equation, giving rise to a ring-shaped image formation made of four highly magnified images, off-centred from the lens centre. Recent observations have revealed new strongly lensed image formations near HU singularities, and many more are expected in ongoing and future observations. Like fold/cusp, image formations near HU also satisfy magnification relation ($R_{\rm hu}$), i.e., the signed magnification sum of the four images equals zero. Here, we study how $R_{\rm hu}$ deviates from zero as a function of area ($A_{\rm hu}$) covered by the image formation near HU and the distance ($d$) of the central maxima image (which is part of the HU image formation) from the lens centre for ideal single- and double-component cluster-scale lenses. For lens ellipticity values $\geq0.3$, the central maxima image will form sufficiently far from the lens centre ($d\gtrsim5''$), similar to the observed HU image formations with $R_{\rm hu}\simeq0$. We also find that, in some cases, double-component and actual cluster-scale lenses can lead to large cross-sections for HU image formations for sources at $z\gtrsim5$, effectively increasing the chances to observe HU image formation at high redshifts. Finally, we study the time delay distribution in the observed HU image formations, finding that not only are these images highly magnified, but the relative time delay between various pairs of HU characteristic image formation has a typical value of $\sim100$ days, an order of magnitude smaller than generic five-image formations in cluster lenses, making such image formations optimal targets for time delay cosmography studies.