Modified gravitational wave propagation with higher modes and its degeneracies with lensing

TL;DR

This paper investigates how modifications in gravitational wave propagation, especially with higher modes, can mimic lensing effects, highlighting the importance of distinguishing these phenomena to test gravity theories with future detectors.

Contribution

It develops an analytical formalism combining WKB and SPA methods for GW propagation with higher modes and explores degeneracies between modified gravity effects and strong lensing.

Findings

WKB and SPA approaches agree when using group velocity.

Degeneracy exists between certain MDR parameters and lensing effects.

Future GW observations must account for these degeneracies to accurately test gravity theories.

Abstract

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.