Phase consistency test to identify type II strongly lensed gravitational wave signals using a single event

Abstract: For gravitationally lensed type II signals, the phase of the dominant (2, 2) mode and the higher order (3, 3) mode is offset by $-\pi/12$. Using this, we develop a test for type II imagery by allowing the phases of the (2,2) and (3,3) modes to vary separately and introducing a new waveform parameter to represent the phase offset between the two. We use simulated, low mass ratio, precessing signals to show that the test can reproduce the $-\pi/12$ phase offset when detected by three detectors for H-L optimal SNR $\gtrsim$ 40 and $\mathcal{M} \leq 30$. We analyze GW190412 and GW190814 using this parameterization, measuring the offset to be $0.13^{+0.22}_{-0.17}$ for GW190412 and $-0.05^{+0.20}_{-0.22}$ for GW190814. We also measure the Bayes Factor in support of no phase offset, $\log_{10} \mathcal{B}_{\Delta \varphi = 0}$, to be $-0.14$ for GW190412 and $0.21$ for GW190814.