Excitation of scalar quasi-normal modes from boson clouds

Abstract

Massive scalar fields on black hole backgrounds generally admit two families of modes: quasi-bound states (QBS) and quasinormal modes (QNM). We demonstrate the orthogonality between the two mode families with respect to a relativistic product. We also find that, although the two families appear on different Riemann sheets of the Green's function of massive scalar perturbations, they can be brought to a single sheet with an appropriate redefinition of the frequency variable. In this variable, it is more natural to see how both mode families can be excited by initial data, and to approximate the Green's function with saddle points. Finally, we investigate the QNM emission from boson clouds - the latter effectively consisting of a single QBS - driven by the tidal perturbation of a second compact object. We show that while the resonant emission of QNMs is generally suppressed, QNM transitions may be more prominent when the interaction with the perturber is non-resonant, such as in the dynamical capture of unbound objects, and when the perturber transits close to the light ring.

Figures (9)

• Figure 1: Complex contour used to compute the product between a QBS and a QNM with positive real frequencies. We place the branch cut of the mode functions between $r_+$ and $r_+ \to +i \infty$. The contour goes around the branch point. In our numerical implementation in Schwarzschild, the vertical paths are placed at ${\rm Re}(r)= r_+ \pm \epsilon$ with $\epsilon=0.1 M$, while the lower horizontal path is at ${\rm Im}(r)=-\epsilon$.
• Figure 2: Product between the fundamental $\ell=m=1$ QNM and the fundamental $\ell=m=1$ QBS as a function of the integral cut-off value $\rm{Im}(r_\infty)$. The red curves are exponential fits, showing convergence to zero. Modes are normalized such that $\langle\langle {\rm QNM, QNM} \rangle\rangle=\langle\langle {\rm QBS, QBS} \rangle\rangle=1$.
• Figure 3: Properties of the Green's function of a massive scalar in the complex plane of the Laplace variable $\omega$. We show the two sheets of the Green's function, arising from the branch cut of the square root $k(\omega)=(\omega^2-\mu^2)^{1/2}$ (red crosses: branch points; our choice for the branch cut: red line). Propagative QNM poles (black crossed circles) lie in the first sheet and have ${\rm Re}(\omega^2)>\mu^2$ (outside the dashed black line), while QBS poles lie in the second sheet and have ${\rm Re}(\omega^2)<\mu^2$. The integration contour runs along and above the real axis in the first sheet.
• Figure 4: Properties of the Green's function in the new complex variable $\eta$, see Eq. \ref{['eq:eta']}. We follow the same conventions as in Fig. \ref{['fig:contour_omega_sheets']}: the solid green line is the original integration contour; the dotted black line marks $|\omega|=\mu$; black crossed circles mark the poles; the red cross marks the Green's function's remaining branch point at $\omega=0$; black lines denote the old axis ${\rm Re}\,\omega=0$ and ${\rm Im}\,\omega=0$. The background color corresponds to the sign of the real part of $\omega$ (red for positive, blue for negative).
• Figure 5: Location of the saddle points (black dots) and steepest-descent contours (dashed green lines) of the massive scalar Green's function in Schwarzschild in the complex $\eta$ plane, at large $t+r'_*>r_*$, with $r_*$ large and $r'_*$ approaching the horizon. See previous Fig. \ref{['fig:contour_other']} for conventions.