Formalism for testing theories of gravity using lensing by compact objects. III. Braneworld gravity

Braneworld gravity is a model that endows physical space with an extra dimension. In the type II Randall-Sundrum braneworld gravity model, the extra dimension modifies the spacetime geometry around black holes, and changes predictions for the formation and survival of primordial black holes. We develop a comprehensive analytical formalism for far-field black hole lensing in this model, using invariant quantities to compute all the geometric optics lensing observables: bending angle, image position, magnification, centroid, and time delay. We then make the first analysis of wave optics in braneworld lensing, working in the semiclassical limit. Through quantitative examples we show that wave optics offers the only realistic way to observe braneworld effects in black hole lensing. We point out that if primordial braneworld black holes exist, have mass $M_{•}$, and contribute a fraction $f_{\mathrm{bh}}$ of the dark matter, then roughly $\sim 3\times {10}^5\times f_{\mathrm{bh}}(M_{•}/{10}^{-18}{M}_{\odot }{)}^{-1}$ of them lie within our Solar System. These objects, which we call “attolenses,” would produce interference fringes in the energy spectra of gamma-ray bursts at energies $E\sim 100(M_{•}/{10}^{-18}{M}_{\odot }{)}^{-1}\mathrm{MeV}$ (which will soon be accessible with the GLAST satellite). Primordial braneworld black holes spread throughout the Universe could produce similar interference effects. If they contribute a fraction ${\Omega }_{•}$ of the total energy density, the probability that gamma-ray bursts are “attolensed” is at least $\sim 0.1{\Omega }_{•}$. If observed, attolensing interference fringes would yield a simple upper limit on $M_{•}$. Detection of a primordial black hole with $M_{•}\lesssim {10}^{-19}{M}_{\odot }$ would challenge general relativity and favor the braneworld model. Further work on lensing tests of braneworld gravity must proceed into the physical optics regime, which awaits a description of the full spacetime geometry around braneworld black holes.