ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O
tokens: to better than 1/ ω for values of ω with a significant spectral component. If Δ l is by hypothesis less than the duration of the undispersed pulse, τ , which is of the order of 1 ms, then the spread in the lensing delays is only of the order of ( R s / R E ) 2 Δ l ( R s / R E ) 2 τ , where R s and R E are respectively the apparent source radius and the Einstein radius of the lens. (That the delay is only second-order in ( R s / R E ) follows from Fermat’s principle, namely that the propagation time is a local minimum relative to all other neighboring trajectories, e.g., Narayan Bartleman 1996 ). If FRBs are at cosmological distances, then, as the apparent source size R S is limited to the FRB pulse width (∼10 −3 c s), and R E 2 is of the order of , it follows that the spread in delays, which is of the order of , is indeed much less than 1 ns. Similarly, the difference in the dispersion time t d ∼ 1 along the two light paths, if the intergalactic medium is smooth, of the order of δθt d , where δθ , the angular separation of the images, obeys δθ 2 / H ∼ 2 Δ l . If Δ l ≪ 1 ms, then δθ ≲ 10 −11 . Conceivably, the intervening medium is clumpy on a much smaller scale than c / H , so the difference in dispersions—in this case governed by the sharp density gradients of the clumps—is larger. This would not destroy the effect; the delay is nonetheless well-defined. It would be due to dispersion differences rather than the lens, but the existence of the multiple paths—and hence the existence of the lens—would be demonstrated. A related question is whether the delay is constant over the course of the FRB. While the apparent source size of an FRB is only about 1 milli-lightsecond, the bulk Lorentz factor Γ of the emitting region must be at least of the order of 10 3.5 (Lyubarsky 2014 ), so the emitting region may move relative to the observer with an apparent velocity of ≳10 3.5 c . Now, for a source at a cosmological distance c / H , the Einstein radius of a lens of mass m is [( c / H )(2 Gm / c 2 )] 1/2 ≃ [10 16 ( m / M ⊙ )] 1/2 cm. So the apparent time to cross the Einstein radius is ∼10 2 [10 3.5 /Γ] [ m / M ⊙ ] 1/2 s, and over the 1 ms duration the change in the source position could be at most 10 −5 [10 3.5 /Γ] [ m / M ⊙ ] −1/2 R E . The change in the delay Δ l due to the motion of the emitting region would then be less than 10 −10 [10 3.5 /Γ] −2 [[ m / M ⊙ ] −1 Δ l ≃ 10 −15 [Γ/10 3.5 ] 2 s, which, for imaginable values of Γ, is much less than the baseband period. The importance of condition ( 3 ) over the full frequency range of significant G ( ω ), together with the broadband requirement ( 7 ) on G , is emphasized. The effect suggested here is not apparent in any formalism that assumes a monochromatic wave because adding a monochromatic wave to a delayed version of itself merely shifts the phase, and changes the amplitude by a modest factor (e.g., Deguchi Watson 1986 , Stanek et al. 2016 ). This is insufficient to reveal to the observer