  Photon rings are key targets for near-future space-based very-long baseline
interferometry missions. The ratio of flux measured between successive
light-rings is characterized by the Lyapunov exponents of the corresponding
nearly-bound null geodesics. Therefore, understanding Lyapunov exponents in
this environment is of crucial importance to understanding black hole
observations in general, and in particular, they may offer a route for
constraining modified theories of gravity. While recent work has made
significant progress in describing these geodesics for Kerr, a theory-agnostic
description is complicated by the fact that Lyapunov exponents are
time-parameterization dependent, which necessitates care when comparing these
exponents in two different theories. In this work, we present a robust
numerical framework for computing and comparing the Lyapunov exponents of null
geodesics in Kerr with those in an arbitrary modified theory. We then present
results obtained from calculating the Lyapunov exponents for null geodesics in
two particular effective theories, scalar Gauss-Bonnet gravity and dynamical
Chern-Simons gravity. Using this framework, we determine accuracy lower-bounds
required before a very-long baseline interferometry observation can constrain
these theories.

General Relativity and Quantum Cosmology