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Abstract

Theories beyond general relativity (GR) modify the propagation of
gravitational waves (GWs). In some, inhomogeneities (aka. gravitational lenses)
allow interactions between the metric and additional fields to cause
lens-induced birefringence (LIB): a different speed of the two linear GW
polarisations ($+$ and $\times$). Inhomogeneities then act as non-isotropic
crystals, splitting the GW signal into two components whose relative time delay
depends on the theory and lens parameters. Here we study the observational
prospects for GW scrambling, i.e when the time delay between both GW
polarisations is smaller than the signal's duration and the waveform recorded
by a detector is distorted. We analyze the latest LIGO-Virgo-KAGRA catalog,
GWTC-3, and find no conclusive evidence for LIB. The highest log Bayes factor
that we find in favour of LIB is $3.21$ for GW$190521$, a particularly loud but
short event. However, when accounting for false alarms due to (Gaussian) noise
fluctuations, this evidence is below 1-$\sigma$. The tightest constraint on the
time delay is $<0.51$ ms (90% C.L.) from GW$200311\_115853$. From the
non-observation of GW scrambling, we constrain the optical depth for LIB,
accounting for the chance of randomly distributed lenses (eg. galaxies) along
the line of sight. Our LIB constraints on a (quartic) scalar-tensor Horndeski
theory are more stringent than solar system tests for a wide parameter range
and comparable to GW170817 in some limits. Interpreting GW190521 as an AGN
binary (i.e. taking an AGN flare as a counterpart) allows even more stringent
constraints. Our results demonstrate the potential and high sensitivity
achievable by tests of GR, based on GW lensing.