Gravitational Lensing by Phase Sheet Folds

Collisionless cold matter in a Hubble flow occupies a 3D sheet in 6D phase space. Perturbations distort the sheet and nonlinear collapse leads to wraps and contortions, leaving the sheet intact with its original local phase density. Fold catastrophes in the projection of this sheet onto the plane of the sky produce line singularities in surface density which can lead to observable gravitational lensing. The universal properties (image positions, magnifications, convergence and shear) of the lensing for a small source near a projected fold are calculated here up to a single scaling parameter for an isolated fold and two parameters for a fold with a smooth background. Estimates are made of scales and lensing probabilities expected for folds in particle dark matter halos producing stellar microlensing events and quasar millilensing events.