Modified gravitational wave propagation with higher modes and its
  degeneracies with lensing
  Low-energy alternatives to General Relativity (GR) generically modify the
phase of gravitational waves (GWs) during their propagation. As detector
sensitivities increase, it becomes key to understand how these modifications
affect the GW higher modes and to disentangle possible degeneracies with
astrophysical phenomena. We apply a general formalism -- the WKB approach --
for solving analytically wave propagation in the spatial domain with a modified
dispersion relation (MDR). We compare this WKB approach to applying a
stationary phase approximation (SPA) in the temporal domain with time delays
associated to the group or particle velocity. To this end, we extend the SPA to
generic signals with higher modes, keeping careful track of reference phases
and arrival times. We find that the WKB approach coincides with the SPA using
the group velocity, in agreement with the principles of wave propagation. We
then explore the degeneracies between a GW propagation with an MDR and a
strongly-lensed GW in GR, since the latter can introduce a
frequency-independent phase shift which is not degenerate with source
parameters in the presence of higher modes. We find that for a particular MDR
there is an exact degeneracy for wave propagation, unlike with the SPA for
particle propagation. For the other cases, we search for the values of the MDR
parameters that minimize the $\chi^2$ and conclude that strongly-lensed GR GWs
could be misinterpreted as GWs in modified gravity. Future MDR constraints with
higher mode GWs should include the possibility of frequency-independent phase
shifts, allowing for the identification of modified gravity and strong lensing
distortions at the same time.
