## Author(s)

Datta, Sayak, Brito, Richard, Hughes, Scott A., Klinger, Talya, Pani, Paolo## Abstract

Tidal heating in a binary black hole system is driven by the absorption of energy and angular momentum by the black hole's horizon. Previous works have shown that this phenomenon becomes particularly significant during the late stages of an extreme mass ratio inspiral (EMRI) into a rapidly spinning massive black hole, a key focus for future low-frequency gravitational-wave observations by (for instance) the LISA mission. Past analyses have largely focused on quasi-circular inspiral geometry, with some of the most detailed studies looking at equatorial cases. Though useful for illustrating the physical principles, this limit is not very realistic astrophysically, since the population of EMRI events is expected to arise from compact objects scattered onto relativistic orbits in galactic centers through many-body events. In this work, we extend those results by studying the importance of tidal heating in equatorial EMRIs with generic eccentricities. Our results suggest that accurate modeling of tidal heating is crucial to prevent significant dephasing and systematic errors in EMRI parameter estimation. We examine a phenomenological model for EMRIs around exotic compact objects by parameterizing deviations from the black hole picture in terms of the fraction of radiation absorbed compared to the BH case. Based on a mismatch calculation we find that reflectivities as small as $|\mathcal{R}|^2 \sim \mathcal{O}(10^{-5})$ are distinguishable from the BH case, irrespective of the value of the eccentricity. We stress, however, that this finding should be corroborated by future parameter estimation studies.

## Figures

Pictorial view of the reflectivity coefficient ${\cal R}$ to quantify the level of absorption of a compact object. Left panel: Since a BH is a perfect absorber, gravitational waves impinging from the right near the horizon are absent on the left. From a one-dimensional radial perspective, there are no reflected waves, hence ${\cal R}=0$. Right panel: an object whose interaction with gravitational waves is negligible (e.g. a perfect-fluid star) has zero absorption. Perhaps counter-intuitively, from a one-dimensional radial perspective, such an object reflects all impinging radiation, so ${\cal R}=1$. Middle panel: intermediate situation in which the object is only partially absorbing.

Maximum change to the inspiral trajectory arising due to tidal heating. In these four panels, we examine a variety of prograde inspirals; in all cases, the larger BH has mass $10^6\,M_\odot$, and the system has mass ratio $M/\mu = 3\times10^4$. We vary the spin $\chi$ of the larger BH (horizontal axis of the figures), as well as the initial eccentricity $e_i$. We choose the initial semi-latus rectum $p_i$ such that the orbital harmonic $\omega_{20} = 2\pi \times 10^{-3}\,{\rm sec}^{-1}$. (The $m = 2$, $n = 0$ harmonic typically makes the loudest contribution to the gravitational waveform.) The four panels compare inspirals computed in ``normal'' GR ($\mathcal{R} = 0$) with maximum reflectivity ($\mathcal{R} = 1$); the various changes we compute scale with the value of $|\mathcal{R}|^2$. Top left shows how the duration (which varies from $0.8$ months to 81.9 months across this parameter space; our ``month'' is 30 days) changes with reflectivity; top right shows similar information, but presented as a fraction of the total inspiral duration. Both panels indicate that, across much of the parameter space, tidal heating makes a non-negligible change to the inspiral duration, just as previous work indicated for quasi-circular inspiral. The bottom panels quantify the extent to which the path in the $(p,e)$ plane is affected by tidal heating, with bottom left showing how the value of $p$ at which inspiral ends changes, and bottom right showing similar data for the final value of $e$. The change to the trajectory is a qualitatively new aspect to the problem seen in the eccentric case. By contrast, for quasi-circular inspiral, all systems evolve with $e = 0$ through a fixed range of orbital radius. This illustrates how the physics of these systems are complicated when we consider more realistic orbital configurations.

Maximum change to the inspiral trajectory arising due to tidal heating. In these four panels, we examine a variety of prograde inspirals; in all cases, the larger BH has mass $10^6\,M_\odot$, and the system has mass ratio $M/\mu = 3\times10^4$. We vary the spin $\chi$ of the larger BH (horizontal axis of the figures), as well as the initial eccentricity $e_i$. We choose the initial semi-latus rectum $p_i$ such that the orbital harmonic $\omega_{20} = 2\pi \times 10^{-3}\,{\rm sec}^{-1}$. (The $m = 2$, $n = 0$ harmonic typically makes the loudest contribution to the gravitational waveform.) The four panels compare inspirals computed in ``normal'' GR ($\mathcal{R} = 0$) with maximum reflectivity ($\mathcal{R} = 1$); the various changes we compute scale with the value of $|\mathcal{R}|^2$. Top left shows how the duration (which varies from $0.8$ months to 81.9 months across this parameter space; our ``month'' is 30 days) changes with reflectivity; top right shows similar information, but presented as a fraction of the total inspiral duration. Both panels indicate that, across much of the parameter space, tidal heating makes a non-negligible change to the inspiral duration, just as previous work indicated for quasi-circular inspiral. The bottom panels quantify the extent to which the path in the $(p,e)$ plane is affected by tidal heating, with bottom left showing how the value of $p$ at which inspiral ends changes, and bottom right showing similar data for the final value of $e$. The change to the trajectory is a qualitatively new aspect to the problem seen in the eccentric case. By contrast, for quasi-circular inspiral, all systems evolve with $e = 0$ through a fixed range of orbital radius. This illustrates how the physics of these systems are complicated when we consider more realistic orbital configurations.

Maximum change to the inspiral trajectory arising due to tidal heating. In these four panels, we examine a variety of prograde inspirals; in all cases, the larger BH has mass $10^6\,M_\odot$, and the system has mass ratio $M/\mu = 3\times10^4$. We vary the spin $\chi$ of the larger BH (horizontal axis of the figures), as well as the initial eccentricity $e_i$. We choose the initial semi-latus rectum $p_i$ such that the orbital harmonic $\omega_{20} = 2\pi \times 10^{-3}\,{\rm sec}^{-1}$. (The $m = 2$, $n = 0$ harmonic typically makes the loudest contribution to the gravitational waveform.) The four panels compare inspirals computed in ``normal'' GR ($\mathcal{R} = 0$) with maximum reflectivity ($\mathcal{R} = 1$); the various changes we compute scale with the value of $|\mathcal{R}|^2$. Top left shows how the duration (which varies from $0.8$ months to 81.9 months across this parameter space; our ``month'' is 30 days) changes with reflectivity; top right shows similar information, but presented as a fraction of the total inspiral duration. Both panels indicate that, across much of the parameter space, tidal heating makes a non-negligible change to the inspiral duration, just as previous work indicated for quasi-circular inspiral. The bottom panels quantify the extent to which the path in the $(p,e)$ plane is affected by tidal heating, with bottom left showing how the value of $p$ at which inspiral ends changes, and bottom right showing similar data for the final value of $e$. The change to the trajectory is a qualitatively new aspect to the problem seen in the eccentric case. By contrast, for quasi-circular inspiral, all systems evolve with $e = 0$ through a fixed range of orbital radius. This illustrates how the physics of these systems are complicated when we consider more realistic orbital configurations.

Maximum change to the inspiral trajectory arising due to tidal heating. This figure shows the same information as Fig.\ \ref{fig:orbit change pro}, but now considers retrograde orbits. The total duration of inspirals in this case varies from 8.8 to 55.3 months across this parameter space. Although similar trends are seen as those found in the prograde examples, the changes due to tidal heating are typically much smaller here. This is not surprising: the last stable orbit is typically at much wider separation for retrograde orbits, so the influence of tidal heating (which is strongest as orbits come close to the event horizon) tends to be smaller for these orbits.

Maximum change to the inspiral trajectory arising due to tidal heating. This figure shows the same information as Fig.\ \ref{fig:orbit change pro}, but now considers retrograde orbits. The total duration of inspirals in this case varies from 8.8 to 55.3 months across this parameter space. Although similar trends are seen as those found in the prograde examples, the changes due to tidal heating are typically much smaller here. This is not surprising: the last stable orbit is typically at much wider separation for retrograde orbits, so the influence of tidal heating (which is strongest as orbits come close to the event horizon) tends to be smaller for these orbits.

Maximum change to the inspiral trajectory arising due to tidal heating. This figure shows the same information as Fig.\ \ref{fig:orbit change pro}, but now considers retrograde orbits. The total duration of inspirals in this case varies from 8.8 to 55.3 months across this parameter space. Although similar trends are seen as those found in the prograde examples, the changes due to tidal heating are typically much smaller here. This is not surprising: the last stable orbit is typically at much wider separation for retrograde orbits, so the influence of tidal heating (which is strongest as orbits come close to the event horizon) tends to be smaller for these orbits.

Two moments along an example waveform, computed for three different reflectivities. Both panels show waveforms corresponding to a prograde inspiral into a BH with $\chi = 0.9$, for a system with initial eccentricity $e = 0.8$; other parameters are as described in the caption to Fig.\ \ref{fig:orbit change pro}. The left panel shows a span of $0.001$ month, or roughly 45 minutes starting 2 months into inspiral; the right panel shows the same thing, but 3 months into inspiral. Notice that at two months, the case with $|\mathcal{R}|^2 = 0.5$ is highly dephased from the other two examples; after three months, even the case with $|\mathcal{R}|^2 = 10^{-3}$ is noticeably dephased from the GR waveform ($\mathcal{R} = 0$).

Two moments along an example waveform, computed for three different reflectivities. Both panels show waveforms corresponding to a prograde inspiral into a BH with $\chi = 0.9$, for a system with initial eccentricity $e = 0.8$; other parameters are as described in the caption to Fig.\ \ref{fig:orbit change pro}. The left panel shows a span of $0.001$ month, or roughly 45 minutes starting 2 months into inspiral; the right panel shows the same thing, but 3 months into inspiral. Notice that at two months, the case with $|\mathcal{R}|^2 = 0.5$ is highly dephased from the other two examples; after three months, even the case with $|\mathcal{R}|^2 = 10^{-3}$ is noticeably dephased from the GR waveform ($\mathcal{R} = 0$).

Total dephasing found as a function of $\chi$ and $e_i$ for prograde inspirals, comparing the ``normal'' GR inspiral ($\mathcal{R} = 0$) to maximum reflectivity ($\mathcal{R} = 1$). The binaries considered are the same ones described in the caption to Fig.\ \ref{fig:orbit change pro}. Left panel shows the change to the accumulated azimuthal phase, $\delta\Phi_\phi$; right panel shows the change to the radial phase $\delta\Phi_r$. Over a very large section of this parameter space, the change is quite large (dozens to thousands of radians). Indeed, only within the narrow white band to the left of both panels do we have $|\delta\Phi| \le 1$. It is worth noting that these dephasings scale proportional to $|\mathcal{R}|^2$, and inversely proportional to the system's mass ratio.

Total dephasing found as a function of $\chi$ and $e_i$ for prograde inspirals, comparing the ``normal'' GR inspiral ($\mathcal{R} = 0$) to maximum reflectivity ($\mathcal{R} = 1$). The binaries considered are the same ones described in the caption to Fig.\ \ref{fig:orbit change pro}. Left panel shows the change to the accumulated azimuthal phase, $\delta\Phi_\phi$; right panel shows the change to the radial phase $\delta\Phi_r$. Over a very large section of this parameter space, the change is quite large (dozens to thousands of radians). Indeed, only within the narrow white band to the left of both panels do we have $|\delta\Phi| \le 1$. It is worth noting that these dephasings scale proportional to $|\mathcal{R}|^2$, and inversely proportional to the system's mass ratio.

Same as Fig.\ \ref{fig: dephasing pro} but for retrograde inspirals. In contrast to prograde configurations, there is no place in this parameter space where the net dephasing is zero; we find non-negligible dephasing for all retrograde configurations.

Same as Fig.\ \ref{fig: dephasing pro} but for retrograde inspirals. In contrast to prograde configurations, there is no place in this parameter space where the net dephasing is zero; we find non-negligible dephasing for all retrograde configurations.

Mismatch $\mathcal{M} = 1 - \mathcal{O}$ for prograde inspiral as a function of observation time between the plus polarization for a waveform computed with $|\mathcal{R}|^2 \neq 0$ (ECO) and another computed with $|\mathcal{R}|^2 = 0$ (BH), maximized over time and phase shift. Each panel shows a different starting eccentricity; in all cases, the large BH has spin $\chi = 0.9$. All other parameters are as described in the caption to Fig.\ \ref{fig:orbit change pro}. The dashed horizontal line shows the threshold $\mathcal{M} = 1/(2\rho^2)$, with $\rho = 20$ chosen as a fiducial SNR associated with the true signal. Notice that inspirals with reflectivity parameter $|\mathcal{R}|^2 = 10^{-3}$ exceed this threshold in all cases we consider here.

Mismatch $\mathcal{M} = 1 - \mathcal{O}$ for prograde inspiral as a function of observation time between the plus polarization for a waveform computed with $|\mathcal{R}|^2 \neq 0$ (ECO) and another computed with $|\mathcal{R}|^2 = 0$ (BH), maximized over time and phase shift. Each panel shows a different starting eccentricity; in all cases, the large BH has spin $\chi = 0.9$. All other parameters are as described in the caption to Fig.\ \ref{fig:orbit change pro}. The dashed horizontal line shows the threshold $\mathcal{M} = 1/(2\rho^2)$, with $\rho = 20$ chosen as a fiducial SNR associated with the true signal. Notice that inspirals with reflectivity parameter $|\mathcal{R}|^2 = 10^{-3}$ exceed this threshold in all cases we consider here.

Mismatch $\mathcal{M} = 1 - \mathcal{O}$ for prograde inspiral as a function of observation time between the plus polarization for a waveform computed with $|\mathcal{R}|^2 \neq 0$ (ECO) and another computed with $|\mathcal{R}|^2 = 0$ (BH), maximized over time and phase shift. Each panel shows a different starting eccentricity; in all cases, the large BH has spin $\chi = 0.9$. All other parameters are as described in the caption to Fig.\ \ref{fig:orbit change pro}. The dashed horizontal line shows the threshold $\mathcal{M} = 1/(2\rho^2)$, with $\rho = 20$ chosen as a fiducial SNR associated with the true signal. Notice that inspirals with reflectivity parameter $|\mathcal{R}|^2 = 10^{-3}$ exceed this threshold in all cases we consider here.

The characteristics of inspiral when the central object is assumed to be a BH, i.e., when we put $\mathcal{R} = 0$. Each panel shows some important characteristic of the inspiral for a system that begins with spin $\chi$ (horizontal axis) and starting eccentricity $e_i$ (vertical axis), and for which $M = 10^6\,M_\odot$, $M/\mu = 3\times10^4$. All inspirals begin when the GW frequency corresponding to the $m = 2$, $n = 0$ voice crosses $10^{-3}$ Hz. Left-hand panels show results for prograde inspirals; right-hand panels are for retrograde. From top to bottom, the panels show the final value of semi-latus rectum, $p_f$, when inspiral ends; the final value of eccentricity $e_f$; and the total inspiral duration in months. Note that our retrograde data covers a smaller span of initial eccentricity than the prograde data, since high eccentricity retrograde inspirals have very short duration with the above constraints.

The characteristics of inspiral when the central object is assumed to be a BH, i.e., when we put $\mathcal{R} = 0$. Each panel shows some important characteristic of the inspiral for a system that begins with spin $\chi$ (horizontal axis) and starting eccentricity $e_i$ (vertical axis), and for which $M = 10^6\,M_\odot$, $M/\mu = 3\times10^4$. All inspirals begin when the GW frequency corresponding to the $m = 2$, $n = 0$ voice crosses $10^{-3}$ Hz. Left-hand panels show results for prograde inspirals; right-hand panels are for retrograde. From top to bottom, the panels show the final value of semi-latus rectum, $p_f$, when inspiral ends; the final value of eccentricity $e_f$; and the total inspiral duration in months. Note that our retrograde data covers a smaller span of initial eccentricity than the prograde data, since high eccentricity retrograde inspirals have very short duration with the above constraints.

The characteristics of inspiral when the central object is assumed to be a BH, i.e., when we put $\mathcal{R} = 0$. Each panel shows some important characteristic of the inspiral for a system that begins with spin $\chi$ (horizontal axis) and starting eccentricity $e_i$ (vertical axis), and for which $M = 10^6\,M_\odot$, $M/\mu = 3\times10^4$. All inspirals begin when the GW frequency corresponding to the $m = 2$, $n = 0$ voice crosses $10^{-3}$ Hz. Left-hand panels show results for prograde inspirals; right-hand panels are for retrograde. From top to bottom, the panels show the final value of semi-latus rectum, $p_f$, when inspiral ends; the final value of eccentricity $e_f$; and the total inspiral duration in months. Note that our retrograde data covers a smaller span of initial eccentricity than the prograde data, since high eccentricity retrograde inspirals have very short duration with the above constraints.

The absolute change in final eccentricity $e_f$ (left-hand panels) and final semi-latus rectum $p_f$ (right) comparing a pure GR inspiral ($\mathcal{R} = 0$) with maximum reflectivity ($\mathcal{R} = 1$). Top is for prograde inspiral; bottom is for retrograde. These plots complement the fractional changes shown in Figs.\ \ref{fig:orbit change pro} and \ref{fig: orbit change ret}.

The absolute change in final eccentricity $e_f$ (left-hand panels) and final semi-latus rectum $p_f$ (right) comparing a pure GR inspiral ($\mathcal{R} = 0$) with maximum reflectivity ($\mathcal{R} = 1$). Top is for prograde inspiral; bottom is for retrograde. These plots complement the fractional changes shown in Figs.\ \ref{fig:orbit change pro} and \ref{fig: orbit change ret}.

The absolute change in final eccentricity $e_f$ (left-hand panels) and final semi-latus rectum $p_f$ (right) comparing a pure GR inspiral ($\mathcal{R} = 0$) with maximum reflectivity ($\mathcal{R} = 1$). Top is for prograde inspiral; bottom is for retrograde. These plots complement the fractional changes shown in Figs.\ \ref{fig:orbit change pro} and \ref{fig: orbit change ret}.

Mismatch $\mathcal{M} = 1 - \mathcal{O}$ for retrograde inspiral; otherwise the same as Fig.\ \ref{fig:mismatch}. Notice that, just as in the prograde examples, inspirals with reflectivity parameter $|\mathcal{R}|^2 = 10^{-3}$ exceed the $10^{-3}$ mismatch threshold in all cases we consider here.

Mismatch $\mathcal{M} = 1 - \mathcal{O}$ for retrograde inspiral; otherwise the same as Fig.\ \ref{fig:mismatch}. Notice that, just as in the prograde examples, inspirals with reflectivity parameter $|\mathcal{R}|^2 = 10^{-3}$ exceed the $10^{-3}$ mismatch threshold in all cases we consider here.

Mismatch $\mathcal{M} = 1 - \mathcal{O}$ for retrograde inspiral; otherwise the same as Fig.\ \ref{fig:mismatch}. Notice that, just as in the prograde examples, inspirals with reflectivity parameter $|\mathcal{R}|^2 = 10^{-3}$ exceed the $10^{-3}$ mismatch threshold in all cases we consider here.

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