Testing Einstein’s theory with lensing of gravitational waves

A new framework for gravitational wave propagation in general space-times and non-standard gravity theories

December 21, 2020

Alternatives to Einstein's theory predict differences in how gravitational waves propagate through the Universe. Previous studies considered such propagation effects on a homogeneous universe, representing the average over all directions. The authors developed a general framework to study gravitational waves propagating through gravitational lenses (galaxies or other matter structures), unveiling novel effects that can be used to test gravity with unprecedented precision. The growing number of gravitational-wave signals detected by LIGO and Virgo increases the chance of finding a close source-lens alignment, and thus the likelihood of performing these novel tests.

Paper abstract

Schematic diagram of the gravitational wave lensing beyond general relativity. A gravitational wave emitted by a binary black hole splits into its propagation eigenstates (wave-forms in color) when it enters the region enclosed by r∗ where modify gravity backgrounds are relevant (note that in general this scale can be different from the scale of strong lensing, i.e. the Einstein radius rE). Depending of the time delays between the propagation eigenstates the signal detected could be scrambled or echoed. If the gravitational wave travels closer than the Einstein radius, multiple images could be formed as indicated by the gray solid trajectories. — Schematic diagram of the gravitational wave lensing beyond general relativity. A gravitational wave emitted by a binary black hole splits into its propagation eigenstates (wave-forms in color) when it enters the region enclosed by r_∗ where modify gravity backgrounds are relevant (note that in general this scale can be different from the scale of strong lensing, i.e. the Einstein radius *r_E*). Depending of the time delays between the propagation eigenstates the signal detected could be scrambled or echoed. If the gravitational wave travels closer than the Einstein radius, multiple images could be formed as indicated by the gray solid trajectories.

Schematic diagram of the gravitational wave lensing beyond general relativity. A gravitational wave emitted by a binary black hole splits into its propagation eigenstates (wave-forms in color) when it enters the region enclosed by r_∗ where modify gravity backgrounds are relevant (note that in general this scale can be different from the scale of strong lensing, i.e. the Einstein radius *r_E*). Depending of the time delays between the propagation eigenstates the signal detected could be scrambled or echoed. If the gravitational wave travels closer than the Einstein radius, multiple images could be formed as indicated by the gray solid trajectories.

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_x) 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.