Dynamical quasinormal mode excitation

Abstract

We study the dynamical excitation of quasinormal modes (QNMs) through the plunge, merger and ringdown of an extreme-mass-ratio-inspiral into a Schwarzschild black hole, for generic orbital configurations. We work out the QNM causality condition, crucial to eliminate amplitude divergences and to incorporate horizon redshift effects. We then use it to derive a model of the time-dependent QNM excitation via a Green's function approach, driven by the point-particle source on a given trajectory. Our model predicts that: i) QNMs propagates along hyperboloidal slices in the minimal gauge; ii) the signal is composed of an ``activation'' term, depending on the source past history, and a local ``impulsive'' term; iii) amplitudes grow in time in an ``activation function'' fashion, and the waveform displays a stationary ringdown regime at times $\sim 10-20M$ after its peak; iv) at these late times, an infinite tower of non-oscillatory, exponentially-damped terms appear: the redshift terms. The model is in good agreement with numerical solutions, capturing the main waveform features after the peak. Additional components of the Green's function are required to complement the QNM description and reproduce the plunge-merger waveform. We predict the late-time, stationary amplitude of the quadrupolar mode as a function of eccentricity, in agreement with accurate numerical solutions, marking the first time that QNM amplitudes are predicted for generic binary configurations. Our work provides a first solid step towards analytically modeling the inspiral's imprint onto ringdown signals, generalizable to include higher orders in the mass ratio, black hole spin, non-vacuum configurations and corrections to the Einstein-Hilbert action.

Figures (16)

• Figure 1: Left: Standard contour used in the QNM literature Leaver:1986gdAndersson:1996cm. Right: Smooth deformation of the contour on the left (closed contour, thick line). As the deformation is performed without crossing any singularity, integration along the two closed contours yields the same result. On the left, the radius of the two quarters of circumference is taken to $R\rightarrow\infty$ in the lower-half plane. On the right, we take $R_{\omega}\rightarrow\infty$ along the real line, and include an infinite number of overtones $n\rightarrow\infty$ along the imaginary axis.
• Figure 2: Time $t = t_{\rm obs}$ at which an observer located at $r_{*,\rm obs} = 100$ sees an impulsive signal emitted at $r_*'$ and $t'=0$. In black, a signal traveling on the light-cone $t_{\rm obs}-r_{*,\rm obs}=t'-r_*'$. In blue, a perturbation scattering at $r_*=0$, i.e. $t_{\rm obs}-r_{*,\rm obs}=t'+r_*'$. In purple, the QNMs propagation $t_{\rm obs}-r_{*,\rm obs}=\mathcal{C}(t',r_*')$, with $\mathcal{C}(t',r_*')$ defined in Eq. \ref{['eq:QNMs_causality']}.
• Figure 7: Activation $\psi_{22n\pm}$ contribution rescaled with the leading QN decay $e^{|\omega^{\rm Im}_{22n}|(\tau-\tau_{\rm LR})}$, vs. the retarded time of the observer at $\mathcal{I}^+$, relative to the particle's apparent LR crossing $\tau_{LR}$, Eq. \ref{['eq:tau_LR']}. We show several $(22n+)$ modes (thick lines) and mirror modes $(22n-)$ (dot-dashed), with overtones marked in different colors. On the top horizontal axes the apparent location $\bar{r}$ of the test-particle emitting the signal observed at $\tau$, as defined in Eq. \ref{['eq:causality_trajectory']}. Results relative to a quasi-circular inspiral-plunge (left), $e_0=0.0$ in Table \ref{['tab:sims_ecc']}, and a radial infall (right) from $r_0=50$ with initial energy $E_0=1.00$. Regions in which the curves are constant indicate the time interval for which the QN constant amplitude behavior is dominant in the activation contribution, before being swamped by the redshift terms.
• Figure 10: Left: Leading redshift term, computed by summing the activation and impulsive contributions of both regular and mirror modes up to $N$ overtones, see Eq. \ref{['eq:convergence_redshift_sum']}, vs. the retarded time $\tau$ of the observer at $\mathcal{I}^+$. On the top axis, the apparent location of the test-particle, Eq. \ref{['eq:causality_trajectory']}. In dot-dashed black, the quadrupole $\Psi^{\rm Num}_{22}$, computed numerically with the RWZHyp code. Quantities are rescaled by the fundamental mode decay. Right: Total leading redshift term, summing contributions up to $N=6$ in Eq. \ref{['eq:convergence_redshift_sum']} (pink, solid), and analytical prediction for the quadrupole strain $\Psi_{22}$, subtracting the $n=0$ activation contributions (violet, dashed). Results relative to the quasi-circular inspiral $e_0=0.0$ in Table. \ref{['tab:sims_ecc']}. The redshift predictions are not valid in the shaded region.
• Figure 11: Amplitude (left) and phase (right) of the quadrupole at $\tau=\tau_{\rm LR}+40$ for the eccentric orbits in Table \ref{['tab:sims_ecc']}, as a function of the orbit's impact parameter at LR crossing (bottom $x$-axis) and the eccentricity at the separatrix (top $x$-axis). We show the numerical results obtained with the RWZHyp code (black empty circles) and the analytical prediction computed as in Eq. \ref{['eq:signal_exc_imp_contribs']} (filled dots). Results are rescaled with respect to the quasi-circular (labeled as QC) case.