Defining eccentricity for gravitational wave astronomy (2024)

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  Figure 1

  Eccentricity and mean anomaly measured using the waveform from an equal-mass nonspinning eccentric NR simulation (SXS:BBH:2312 [48, 117]). Left: time evolution of the eccentricity $e_{\mathrm{gw}}$ (upper panel) and frequency of the (2, 2) waveform mode ${\omega }_{22}$ (lower panel). ${\omega }_{22}^{\mathrm{p}}(t)$ and ${\omega }_{22}^{\mathrm{a}}(t)$ are interpolants through ${\omega }_{22}(t)$ evaluated at the pericenters (blue circles) and apocenters (pink squares), respectively. Equation(8) is used to compute $e_{\mathrm{gw}}(t)$ given ${\omega }_{22}^{\mathrm{p}}(t)$ and ${\omega }_{22}^{\mathrm{a}}(t)$. Right: time evolution of the mean anomaly $l_{\mathrm{gw}}$ (upper panel) and ${\omega }_{22}$ (lower panel). The vertical dashed gray lines denote the pericenter times. $l_{\mathrm{gw}}(t)$ grows linearly in time from 0 to $2\pi$ between successive pericenters [Eq.(10)].

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  Figure 2

  Different methods to construct a monotonically increasing frequency to replace ${\omega }_{22}(t)$, in order to set the reference frequency $f_{\mathrm{ref}}$ for eccentric binaries. We consider two different approaches: (i)${\omega }_{22}^{\mathrm{mean}}(t)$, the mean of ${\omega }_{22}^{\mathrm{p}}(t)$ and ${\omega }_{22}^{\mathrm{a}}(t)$, and (ii)$⟨{\omega }_{22}⟩(t)$, an interpolant through the orbit-averaged ${\omega }_{22}$ [Eq.(12)]. We show SEOBNRv4EHM waveforms with three different eccentricities; the binary parameters are given in the figure text. While the two approaches agree for small eccentricities, they deviate significantly at large eccentricities. We adopt $⟨{\omega }_{22}⟩(t)$ as it captures the correct frequency scale in an orbit-averaged sense (Sec.2e).

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  Figure 3

  How to truncate time-domain eccentric waveforms while retaining all frequencies above $f_{\mathrm{low}}=20\mathrm{Hz}$. The orange, blue, and pink curves show different sections of ${\omega }_{22}(t)$ for an eccentric SEOBNRv4EHM waveform (with binary parameters shown in the title). If we discard all times below the point where the orbit-averaged frequency $⟨f_{22}⟩\equiv ⟨{\omega }_{22}⟩/(2\pi )$ (pink dashed curve) crosses $f_{\mathrm{low}}=20\mathrm{Hz}$, only the pink section is retained, and the blue section is discarded even though it contains some frequencies above 20Hz. On the other hand, using $f_{22}^{\mathrm{p}}\equiv {\omega }_{22}^{\mathrm{p}}/(2\pi )$ (blue dashed curve) to pick this time ensures that the discarded region (orange) contains no frequencies above 20Hz.

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  Figure 4

  Limitations of the amplitude and frequency methods in identifying pericenters (blue circles) and apocenters (pink squares) for a low eccentricity waveform. These methods (top two rows) detect only the first few pericenters or apocenters and fail once sufficient eccentricity is radiated away. On the other hand, the residual amplitude and residual frequency methods (bottom two rows) can detect all of the pericenters or apocenters present. The waveform is generated using SEOBNRv4EHM, and the binary parameters are given in the title.

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  Figure 5

  Comparison of the amplitude (top) and the frequency (bottom) of an eccentric SEOBNRv4EHM waveform to those of its quasicircular counterpart. The binary parameters are shown in the figure text. Both waveforms are aligned so that $t=0$ occurs at the peak of $A_{22}$. The quasicircular counterpart captures the secular growth in the amplitude and frequency of the eccentric waveform.

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  Figure 6

  Illustration of the frequency fits method. Left: the blue circles indicate the $2N+1=7$ extrema through which the fitting function Eq.(30) passes. The lower panel shows the envelope-subtracted data from which the extrema $T_{\alpha }$ are determined. The solid blue circle indicates the central extremum, whose parameters are used for the eccentricity definition. The pink square and the pink dashed line show the analogous construction for the apocenter passages. Right: enlargement of the region around the solid markers in the upper panel on the left. The waveform is generated using SEOBNRv4EHM, and the binary parameters are given in the title.

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  Figure 7

  Comparison of $e_{\mathrm{gw}}$ and $e_{{\omega }_{22}}$ to the geodesic eccentricity $e_{\mathrm{geo}}$ in the $q\to \infty$ limit, as a function of the orbit-averaged frequency $⟨{\omega }_{22}⟩$. In the left panel, the colors show the absolute difference between $e_{\mathrm{geo}}$ and $e_{\mathrm{gw}}$ measured using Eq.(8) with the amplitude method. The right panel shows the same for $e_{{\omega }_{22}}$. $e_{\mathrm{geo}}$ is closer to $e_{\mathrm{gw}}$ than $e_{{\omega }_{22}}$ by about 2 orders of magnitude.

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  Figure 8

  Demonstration of the measurement of eccentricity using the gw_eccentricity [93] package for waveforms of different origins: PN, EOB, NR, and EMRI. The binary parameters are indicated in the figure text. In each subplot, the lower panel shows the real part of ${\mathcal{h}}_{22}$, and the upper panel shows the measured eccentricity. We consider three different methods for identifying the pericenters and apocenters: amplitude, residual amplitude, and amplitude fits.

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  Figure 9

  $e_{\mathrm{gw}}$ vs $e_{\mathrm{eob}}$ at the initial time, for SEOBNRv4EHM waveforms with varying $e_{\mathrm{eob}}$, but keeping the other binary parameters fixed (given in figure title). $e_{\mathrm{eob}}$ is the model’s internal eccentricity, specified at $t_0=-20000M$. $e_{\mathrm{gw}}$ is evaluated at its first available time ${\widehat{t}}_0$. We consider three different methods for locating pericenters and apocenters: amplitude, residual amplitude, and amplitude fits. The amplitude method breaks down for small eccentricities ($e_{\mathrm{eob}}\lesssim {10}^{-3}$), while the residual amplitude and amplitude fits method follow the expected $e_{\mathrm{gw}}=e_{\mathrm{eob}}$ trend down to $e_{\mathrm{eob}}={10}^{-5}$.

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  Figure 10

  $e_{\mathrm{gw}}$ vs the internal definition of eccentricity, for waveforms of different origin, for equal-mass nonspinning binaries with varying eccentricity. For the NR waveforms (SpEC), we compute the internal eccentricity at $t_0=1500M$ after the start of the simulation, while for the rest we use $t_0=-20000M$ before peak waveform amplitude. In both cases, $\widehat{t_0}$ is the first available time for $e_{\mathrm{gw}}(t)$. The inset shows the same but on a linear scale and focuses on the $e_{\mathrm{gw}}\le 0.4$ region.

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  Figure 11

  $e_{\mathrm{gw}}(t)$ for SEOBNRv4EHM waveforms with varying $e_{\mathrm{eob}}$ but keeping the other binary parameters fixed (given in the figure title). The method used to locate pericenters and apocenters is indicated in the figure text. The colors indicate the value of $e_{\mathrm{eob}}$, defined at $t_0=-20000M$. The amplitude method breaks down for small eccentricities $e_{\mathrm{gw}}\lesssim {10}^{-3}\dots {10}^{-2}$, especially as one approaches the merger. The residual amplitude and amplitude fits methods continue to compute the eccentricity until $e_{\mathrm{gw}}\sim {10}^{-5}$. The features at $e_{\mathrm{gw}}\sim {10}^{-5}$ arise from the waveform model itself (see Fig.12).

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  Figure 12

  Tracing the noisy features in Fig.11 to the behavior of the SEOBNRv4EHM model at small eccentricities. The top panel shows $e_{\mathrm{gw}}$ for the case with $e_{\mathrm{eob}}=1.05\times {10}^{-5}$ at $t_0=-20000M$, from the middle panel in Fig.11. The bottom panel shows the corresponding $\mathrm{\Delta }{\omega }_{22}$ [Eq.(26)], which helps highlight the modulations due to eccentricity. The drop in $e_{\mathrm{gw}}$ occurs at the same time as an abrupt drop in the eccentricity modulations in $\mathrm{\Delta }{\omega }_{22}$ that arises from a transition function applied to the dynamical variables entering the NQC corrections in SEOBNRv4EHM [46].

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  Figure 13

  Differences in $e_{\mathrm{gw}}(t)$ due to different methods used to locate pericenters and apocenters, for the same SEOBNRv4EHM waveforms as Fig.11. Top left: the curves show $e_{\mathrm{gw}}(t)$ obtained using the residual amplitude method with the quasicircular counterpart also obtained from SEOBNRv4EHM. The colors represent the absolute difference with respect to the $e_{\mathrm{gw}}(t)$ obtained using the amplitude fits method, and the gray region shows the parts where the second method fails to compute $e_{\mathrm{gw}}(t)$. Top right: the same, but now the colors show the difference with respect to the $e_{\mathrm{gw}}(t)$ obtained with residual amplitude method with the quasicircular counterpart obtained from the IMRPhenomT model. In both top panels, the different choices for locating pericenters and apocenters lead to broadly consistent results for $e_{\mathrm{gw}}(t)$, with the only notable differences occurring for (i)small eccentricities ($e_{\mathrm{gw}}\lesssim 5\times {10}^{-3}$) and near the merger, where the SEOBNRv4EHM model also has known issues (see Fig.12), and (ii)large eccentricities ($e_{\mathrm{gw}}\sim 0.9$), where locating apocenters is problematic. The bottom panels show the same as the top panels but when identifying the midpoints between pericenters as apocenters. This leads to more consistent results between different methods, and the largest differences in $e_{\mathrm{gw}}$ decrease by an order of magnitude.

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  Figure 14

  A smoothness test for $e_{\mathrm{gw}}(t)$ at very high eccentricities. The curves show the time evolution of $e_{\mathrm{gw}}$ for SEOBNRv4EHM waveforms with initial eccentricities $0.9\le e_{\mathrm{eob}}\le 0.999$ at $t_0=-5\times {10}^{6}M$. The colors represent $e_{\mathrm{eob}}$ at $t_0$. The binary parameters are shown in the figure title. We use the residual amplitude method to locate pericenters and identify the midpoints between pericenters as apocenters.

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