Signatures of dark and baryonic structures on weakly lensed gravitational waves

Gravitational lensing offers a powerful tool for exploring the matter distribution in the Universe. Thanks to their low frequencies and phase coherence, gravitational waves (GWs) allow for the observation of novel wave-optics features (WOFs) in lensing, inaccessible to electromagnetic signals. Combined with the existing accurate source models, lensed GWs can be used to infer the properties of gravitational lenses. The prospect is particularly compelling for space-borne detectors, where the high signal-to-noise ratio expected from massive black hole binary mergers allows WOFs to be distinguished deep into the weak lensing regime, drastically increasing the detection probability. Here, we investigate in detail the capacity of the LISA mission to detect WOFs caused by dark matter halos, galaxies, and the supermassive black holes (SMBHs) within them. We estimate the total optical depth to be <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>λ</a:mi><a:mrow><a:mi>tot</a:mi></a:mrow></a:msub><a:mo>∼</a:mo><a:mn>6</a:mn><a:mo>×</a:mo><a:msup><a:mn>10</a:mn><a:mrow><a:mo>−</a:mo><a:mn>3</a:mn></a:mrow></a:msup></a:math> for the loudest binaries of total mass <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>M</c:mi><c:mrow><c:mi>BBH</c:mi></c:mrow></c:msub><c:mo>∼</c:mo><c:msup><c:mn>10</c:mn><c:mn>6</c:mn></c:msup><c:msub><c:mi>M</c:mi><c:mo stretchy="false">⊙</c:mo></c:msub></c:math>, with the dominant contribution coming from SMBHs. We also find that WOFs in low-mass binaries <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi>M</f:mi><f:mrow><f:mi>BBH</f:mi></f:mrow></f:msub><f:mo>∼</f:mo><f:msup><f:mn>10</f:mn><f:mn>4</f:mn></f:msup><f:msub><f:mi>M</f:mi><f:mo stretchy="false">⊙</f:mo></f:msub></f:math> are more likely due to the central galaxies. Within our model of gravitational lenses, we predict <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi mathvariant="script">O</i:mi><i:mo stretchy="false">(</i:mo><i:mn>0.1</i:mn><i:mo stretchy="false">)</i:mo><i:mo>−</i:mo><i:mi mathvariant="script">O</i:mi><i:mo stretchy="false">(</i:mo><i:mn>1</i:mn><i:mo stretchy="false">)</i:mo></i:math> weakly lensed events to be detectable during the five years of LISA mission, depending on the source population models. We show that WOFs signatures are very sensitive to the properties of dark matter halos with <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mrow><q:msub><q:mrow><q:mi>M</q:mi></q:mrow><q:mrow><q:mi>vir</q:mi></q:mrow></q:msub><q:mo>∈</q:mo><q:mo stretchy="false">(</q:mo><q:msup><q:mrow><q:mn>10</q:mn></q:mrow><q:mrow><q:mn>6</q:mn></q:mrow></q:msup><q:mi>–</q:mi><q:msup><q:mrow><q:mn>10</q:mn></q:mrow><q:mrow><q:mn>8</q:mn></q:mrow></q:msup><q:mo stretchy="false">)</q:mo><q:msub><q:mrow><q:mi>M</q:mi></q:mrow><q:mrow><q:mo stretchy="false">⊙</q:mo></q:mrow></q:msub></q:mrow></q:math>: Increasing the compactness parameter by <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:mo>∼</v:mo><v:mn>3</v:mn></v:math> in that range raises the detection rate by <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:mo>∼</x:mo><x:mn>26</x:mn></x:math>. Additionally, we show that collective effects from the complex inner halo structure can further enhance detectability. This suggests that lensed GWs in LISA will be an excellent probe of dark-matter theories, baryonic, and halo substructures.