To do this, we did a tree-level calculation (first pair of diagrams immediately below) of the cross-section for gravitational lensing (analagous to Rutherford scattering) in linearized quantum gravity with a cutoff, and then extended the calculation to all orders in perturbation theory (series of diagrams below the first pair). This calculation only converges if we restrict ourselves to the eikonal (geometric-optics) limit, which certainly applies in the astrophysical scenarios of interest.

Such a theory can be derived from a Lagrangian density non-linear in the d’Alembertian.

We also show below the x-ray lensing contours of the lensing system Q0957+661 provided to us by George Chartas for our analysis:

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