Phase effects from strong gravitational lensing of gravitational waves. (arXiv:2008.12814v2 [gr-qc] UPDATED)
<a href="http://arxiv.org/find/gr-qc/1/au:+Ezquiaga_J/0/1/0/all/0/1">Jose María Ezquiaga</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Holz_D/0/1/0/all/0/1">Daniel E. Holz</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Hu_W/0/1/0/all/0/1">Wayne Hu</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Lagos_M/0/1/0/all/0/1">Macarena Lagos</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Wald_R/0/1/0/all/0/1">Robert M. Wald</a>
Assessing the probability that two or more gravitational waves (GWs) are
lensed images of the same source requires an understanding of the image
properties, including their relative phase shifts in strong lensing (SL). For
non-precessing, circular binaries dominated by quadrupole radiation these phase
shifts are degenerate with either a shift in the coalescence phase or a
detector and inclination dependent shift in the orientation angle. This
degeneracy is broken by the presence of higher harmonic modes with $|m|ne 2$
in the former and $|m| ne l$ in the latter. Precession or eccentricity will
also break this degeneracy. This implies that lensed GWs will not necessarily
be consistent with (unlensed) predictions from general relativity (GR).
Therefore, unlike EM lensing, GW SL can lead to images with an observable
modified phase evolution. However, for a wide parameter space, the lensed
waveform is similar enough to an unlensed waveform that detection pipelines
will still find it. For present detectors, templates with a shifted
detector-dependent orientation angle have SNR differences of less than $1%$
for mass ratios up to 0.1, and less than $5%$ for precession parameters up to
0.5 and eccentricities up to 0.4 at 20Hz. The mismatch is lower than $10%$
with the alternative detector-independent coalescence phase shift. Nonetheless,
for a loud enough source, even with one image it may be possible to directly
identify it as a SL image from its non-GR waveform. In more extreme cases,
lensing may lead to considerable distortions, and the lensed GWs may be
undetected with current searches. Nevertheless, an exact template with a phase
shift in Fourier space can always be constructed to fit any lensed GW. We
conclude that an optimal search strategy would incorporate phase information in
all stages, with an exact treatment in the final assessment of the probability
of multiple lensed events.
Assessing the probability that two or more gravitational waves (GWs) are
lensed images of the same source requires an understanding of the image
properties, including their relative phase shifts in strong lensing (SL). For
non-precessing, circular binaries dominated by quadrupole radiation these phase
shifts are degenerate with either a shift in the coalescence phase or a
detector and inclination dependent shift in the orientation angle. This
degeneracy is broken by the presence of higher harmonic modes with $|m|ne 2$
in the former and $|m| ne l$ in the latter. Precession or eccentricity will
also break this degeneracy. This implies that lensed GWs will not necessarily
be consistent with (unlensed) predictions from general relativity (GR).
Therefore, unlike EM lensing, GW SL can lead to images with an observable
modified phase evolution. However, for a wide parameter space, the lensed
waveform is similar enough to an unlensed waveform that detection pipelines
will still find it. For present detectors, templates with a shifted
detector-dependent orientation angle have SNR differences of less than $1%$
for mass ratios up to 0.1, and less than $5%$ for precession parameters up to
0.5 and eccentricities up to 0.4 at 20Hz. The mismatch is lower than $10%$
with the alternative detector-independent coalescence phase shift. Nonetheless,
for a loud enough source, even with one image it may be possible to directly
identify it as a SL image from its non-GR waveform. In more extreme cases,
lensing may lead to considerable distortions, and the lensed GWs may be
undetected with current searches. Nevertheless, an exact template with a phase
shift in Fourier space can always be constructed to fit any lensed GW. We
conclude that an optimal search strategy would incorporate phase information in
all stages, with an exact treatment in the final assessment of the probability
of multiple lensed events.
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