Characterizing the merger of equatorial-eccentric-geodesic plunges in rotating black holes

We study the gravitational waveforms generated by critical, equatorial plunging geodesics of the Kerr metric that start from an unstable-circular-orbit, which describe the test-mass limit of spin-aligned eccentric black hole mergers. The waveforms are generated employing a time-domain eukolsky code. We span different values of the Kerr spin <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo>−</a:mo><a:mn>0.99</a:mn><a:mo>≤</a:mo><a:mi>a</a:mi><a:mo>≤</a:mo><a:mn>0.99</a:mn></a:math> and of the critical eccentricity <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>e</c:mi><c:mi>c</c:mi></c:msub></c:math>, for bound (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mn>0</e:mn><e:mo>≤</e:mo><e:msub><e:mi>e</e:mi><e:mi>c</e:mi></e:msub><e:mo><</e:mo><e:mn>1</e:mn></e:math>) and unbound plunges (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>e</g:mi><g:mi>c</g:mi></g:msub><g:mo>≥</g:mo><g:mn>1</g:mn></g:math>). We find that, contrary to expectations, the waveform modes <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>h</i:mi><i:mrow><i:mo>ℓ</i:mo><i:mi>m</i:mi></i:mrow></i:msub></i:math> do not always manifest a peak for high eccentricities or spins. In case of the dominant <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>h</k:mi><k:mn>22</k:mn></k:msub></k:math> mode, we determine the precise region of the parameter space in which its peak exists. In this region, we provide a characterization of the merger quantities of the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>h</m:mi><m:mn>22</m:mn></m:msub></m:math> mode and of the higher-order modes, providing the merger structure of the equatorial-eccentric plunges of the Kerr spacetime in the test-mass limit.