Environmental effects in extreme mass ratio inspirals: perturbations to the environment in Kerr

Author(s)

Dyson, Conor, Spieksma, Thomas F.M., Brito, Richard, van de Meent, Maarten, Dolan, Sam

Abstract

Future gravitational wave observatories open a unique avenue to study the environments surrounding black holes. Intermediate or extreme mass ratio inspirals will spend thousands to millions of cycles in the sensitivity range of detectors, allowing subtle environmental effects to accumulate in the gravitational waveform. Working in Lorenz gauge and considering equatorial circular orbits, we present the first self-consistent, fully relativistic calculation of a perturbation to a black hole environment due to an inspiraling secondary in the Kerr geometry. As an example case, we consider the environment to be that of a superradiantly grown scalar cloud, though our framework is generalizable to other scenarios. We demonstrate that the scalar field develops a rich wake structure induced by the secondary and compute scalar fluxes emitted to infinity and through the horizon. Relative differences in the fluxes compared to Schwarzschild are tens of percent on large intervals of parameter space, underscoring the importance of modeling in Kerr.

Figures

We show the absolute value of the perturbed scalar field $|\phi^{(1,1)}|$ for $\ell\geq 2$, taking $\alpha = 0.3$, $a = 0.88M$ and $r_{\rm p} = 3.5M$. In the top panel, we show an equatorial slice of the field solution, in which the $\hat{Z}$--axis is aligned with the BH spin. In the bottom panel, we show an azimuthal slice of the field, where the secondary moves ``into the plane.''

We show the absolute value of the perturbed scalar field $|\phi^{(1,1)}|$ for $\ell\geq 2$, taking $\alpha = 0.3$, $a = 0.88M$ and $r_{\rm p} = 3.5M$. In the top panel, we show an equatorial slice of the field solution, in which the $\hat{Z}$--axis is aligned with the BH spin. In the bottom panel, we show an azimuthal slice of the field, where the secondary moves ``into the plane.''


We show the total flux to infinity (solid lines) and through the horizon (dotted lines) considering a prograde orbit and $\alpha = 0.2$ (\emph{left panel}) or $\alpha = 0.3$ (\emph{right panel}). Note that the horizon fluxes are negative on the entire radial domain. The sharp features in the infinity flux in the right panel, computed using Eq.~\eqref{eq:discontinuities}, are marked by vertical dashed lines. Note the Schwarzschild results stop at the innermost stable circular orbit (ISCO) ($r_{\rm p} = 6M$). We sum up to $\ell = 6$ (5) for the infinity (horizon) fluxes.

We show the total flux to infinity (solid lines) and through the horizon (dotted lines) considering a prograde orbit and $\alpha = 0.2$ (\emph{left panel}) or $\alpha = 0.3$ (\emph{right panel}). Note that the horizon fluxes are negative on the entire radial domain. The sharp features in the infinity flux in the right panel, computed using Eq.~\eqref{eq:discontinuities}, are marked by vertical dashed lines. Note the Schwarzschild results stop at the innermost stable circular orbit (ISCO) ($r_{\rm p} = 6M$). We sum up to $\ell = 6$ (5) for the infinity (horizon) fluxes.


We show the relative error $\Delta \mathcal{F}$ in the total flux to infinity (solid lines) and through the horizon (dotted lines), taking our Schwarzschild data as a reference. We consider a prograde orbit and $\alpha = 0.2$ (\emph{left panel}) or $\alpha = 0.3$ (\emph{right panel}). The comparison is between our Kerr and Schwarzschild (red lines) and between our Schwarzschild and the Schwarzschild from an earlier work~\cite{Brito:2023pyl} (purple line), which used a different gauge. In the inset, we show the actual flux $\mathcal{F}^{\mathrm{s}, \infty}$ in Schwarzschild/Kerr from our data (blue/orange solid) compared to the one from~\cite{Brito:2023pyl} (black dashed). The horizontal axis is the same as in the main plot.

We show the relative error $\Delta \mathcal{F}$ in the total flux to infinity (solid lines) and through the horizon (dotted lines), taking our Schwarzschild data as a reference. We consider a prograde orbit and $\alpha = 0.2$ (\emph{left panel}) or $\alpha = 0.3$ (\emph{right panel}). The comparison is between our Kerr and Schwarzschild (red lines) and between our Schwarzschild and the Schwarzschild from an earlier work~\cite{Brito:2023pyl} (purple line), which used a different gauge. In the inset, we show the actual flux $\mathcal{F}^{\mathrm{s}, \infty}$ in Schwarzschild/Kerr from our data (blue/orange solid) compared to the one from~\cite{Brito:2023pyl} (black dashed). The horizontal axis is the same as in the main plot.


We show the fluxes to infinity and the horizon for the scalar and gravitational case, including the correct perturbative prefactors, considering $q = 10^{-6}$, $\epsilon^2 = 0.1\alpha^3$, $\alpha = 0.3$. In the inset, we show the ratio between the two, i.e., $\mathcal{F}_{\rm s}/\mathcal{F}_{\scalebox{0.6}{$\mathrm{GW}$}}$. The horizontal axis is the same as in the main plot.

We show the fluxes to infinity and the horizon for the scalar and gravitational case, including the correct perturbative prefactors, considering $q = 10^{-6}$, $\epsilon^2 = 0.1\alpha^3$, $\alpha = 0.3$. In the inset, we show the ratio between the two, i.e., $\mathcal{F}_{\rm s}/\mathcal{F}_{\scalebox{0.6}{$\mathrm{GW}$}}$. The horizontal axis is the same as in the main plot.


We show the absolute value of the perturbed scalar field $|\phi^{(1,1)}|$ for $\ell\geq 2$, taking $\alpha = 0.3$, $a = 0.88M$. In the top panel, we show an equatorial slice of the field solution at $r_{\rm p} = 41.6M$, in which the $\hat{Z}$--axis is aligned with the BH spin. In the bottom panel, we show an equatorial slice at $r_{\rm p} = 41.8M$.

We show the absolute value of the perturbed scalar field $|\phi^{(1,1)}|$ for $\ell\geq 2$, taking $\alpha = 0.3$, $a = 0.88M$. In the top panel, we show an equatorial slice of the field solution at $r_{\rm p} = 41.6M$, in which the $\hat{Z}$--axis is aligned with the BH spin. In the bottom panel, we show an equatorial slice at $r_{\rm p} = 41.8M$.


We show the contribution from different $\ell$-modes to the flux to infinity in Kerr for $\alpha = 0.3$ with the secondary on a prograde orbit at $r_{\rm p} = 20 M$. Due to selection rules, modes with opposite parity are zero, i.e., when $\ell$ is odd and $m$ is even or vice versa. Additionally, all $m = 0,1$ modes do not contribute to the flux to infinity. The different sized diamonds thus show the contribution from the modes that are not zero, where the higher the $m$, the higher contribution. For example, for $\ell = 8$, we show $m = 8, 6, 4, 2$. Fluxes through the horizon follow a similar trend.

We show the contribution from different $\ell$-modes to the flux to infinity in Kerr for $\alpha = 0.3$ with the secondary on a prograde orbit at $r_{\rm p} = 20 M$. Due to selection rules, modes with opposite parity are zero, i.e., when $\ell$ is odd and $m$ is even or vice versa. Additionally, all $m = 0,1$ modes do not contribute to the flux to infinity. The different sized diamonds thus show the contribution from the modes that are not zero, where the higher the $m$, the higher contribution. For example, for $\ell = 8$, we show $m = 8, 6, 4, 2$. Fluxes through the horizon follow a similar trend.


We show the contribution from different $\ell$-modes of the scalar field perturbation evaluated on the orbital radius of the secondary ($r_{\rm p} = 20 M$). They fall off as with the expected rate, $\ell^{-2}$, indicated by the black dashed line.

We show the contribution from different $\ell$-modes of the scalar field perturbation evaluated on the orbital radius of the secondary ($r_{\rm p} = 20 M$). They fall off as with the expected rate, $\ell^{-2}$, indicated by the black dashed line.


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