Gravitational wave lensing beyond general relativity: birefringence,
  echoes and shadows
  Gravitational waves (GW), as light, are gravitationally lensed by intervening
matter, deflecting their trajectories, delaying their arrival and occasionally
producing multiple images. In theories beyond general relativity (GR), new
gravitational degrees of freedom add an extra layer of complexity and richness
to GW lensing. We develop a formalism to compute GW propagation beyond GR over
general space-times, including kinetic interactions with new fields. Our
framework relies on identifying the dynamical propagation eigenstates (linear
combinations of the metric and additional fields) at leading order in a
short-wave expansion. We determine these eigenstates and the conditions under
which they acquire a different propagation speed around a lens. Differences in
speed between eigenstates cause birefringence phenomena, including time delays
between the metric polarizations (orthogonal superpositions of $h_+,h_\times$)
observable without an electromagnetic counterpart. In particular, GW echoes are
produced when the accumulated delay is larger than the signal's duration, while
shorter time delays produce a scrambling of the wave-form. We also describe the
formation of GW shadows as non-propagating metric components are sourced by the
background of the additional fields around the lens. As an example, we apply
our methodology to quartic Horndeski theories with Vainshtein screening and
show that birefringence effects probe a region of the parameter space
complementary to the constraints from the multi-messenger event GW170817. In
the future, identified strongly lensed GWs and binary black holes merging near
dense environments, such as active galactic nuclei, will fulfill the potential
of these novel tests of gravity.
