Counting Lensed Images: Multiplane Case

Abstract

This chapter extends the work of Chapter 11 to a study of lensed-image counting due to gravitational lenses on any finite number of lens planes (Problem M1). As in Chapter 11, we follow the approach in [Pet95a], employing Morse theory to investigate the number of lensed images of different types. In particular, lensed images are counted within a compact region D not a priori required to contain the whole deflector on each lens plane. The associated time delay function T _y must then obey Morse boundary conditions B relative to D before we can apply Morse theory. We shall show that there is an arbitrarily small linear perturbation of T _y in D such that the resulting function satisfies Morse boundary conditions B relative to D and — most importantly — is also a time delay function.

“What is one and one and one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”

“She can’t do addition,” said the Red Queen. Lewis Carroll