Exceptional Lines and Excitation of (Nearly) Double-Pole Quasinormal Modes: A Semi-Analytic Study in the Nariai Black Hole

Abstract

We show that quasinormal modes (QNMs) of a massive scalar field in Kerr-de Sitter and Myers-Perry black holes exhibit an exceptional line (EL), which is a continuous set of exceptional points (EPs) in parameter space, at which two QNM frequencies and their associated solutions coincide. We find that the EL appears in the parameter space spanned by the scalar mass and the black hole spin parameter, and also in the Nariai limit, i.e., $r_{\rm c} - r_{\rm h} \to 0$, where $r_{\rm c}$ and $r_{\rm h}$ denote the radii of the cosmological and black hole horizons, respectively. We analytically study the amplitudes or excitation factors of QNMs near the EL. Such an analytic treatment becomes possible since, in the Nariai limit, the perturbation equation reduces to a wave equation with the Pöschl-Teller (PT) potential. We discuss the destructive excitation of QNMs and the stability of the ringdown near and at the EL. The transient linear growth of QNMs -- a characteristic excitation pattern near an EP or EL -- together with the conditions under which this linear growth dominates the early ringdown, is also studied analytically. Our conditions apply to a broad class of systems that involve the excitation of (nearly) double-pole QNMs.

Figures (14)

• Figure 1: Schematic picture of BH parameter trajectories (black arrows) in a relevant parameter space spanned by, say, $A$ and $B$, involving an EP (left) and an EL (right). The example assumes that the relevant parameter space is restricted to two dimensions.
• Figure 2: Phase space of the MP-dS BH with a single rotation and $M=1$. The spacetime dimension is set to $d=4$ (left), $d=5$ (center), and $d=6$ (right). The black dashed line stands for the upper bound in \ref{['upper_spin_d5']}. The BH solution exists in the gray-shaded region.
• Figure 3: ( Left) Distribution of QNMs for prograde and retrograde modes when $a=0$, $\mu=0.18$, and $k=j=m=0$ up to $n=7$ modes. ( Right) Trajectories of the fundamental QNMs, $\omega_0^{\pm}$, near or at the exceptional point. The value $\mu$ decreases from $0.18$ to $0.16$ along the arrows, and the two QNMs intersect at $\mu=1/6$.
• Figure 4: ( Upper panels) Ringdown waveform $|h_{\rm G}|$ (black solid) and the ringdown reconstruction $|h_{\rm E}|$ with $n_{\rm max} = 2$ (red dashed), $n_{\rm max} = 6$ (blue dot-dashed) and $n_{\rm max} = 20$ (green dotted) for (a) $\mu=1/6-3\times10^{-4}$ and (b) $\mu=1/6-10^{-10}$, where $N_+ = N_- \eqcolon n_{\rm max}$. ( Bottom left) The excitation factors $B_0^{\pm}$ (squares and circles) are calculated with the range of $0.166\leq \mu \leq 0.167$ in the step size of $10^{-10}$. ( Bottom right) The absolute values of the excitation factor $|B_0^{\pm}|$ of both the prograde and retrograde mode. The excitation factor diverges at $\mu=1/6$ (gray-dashed line), where the double-pole QNM appears.
• Figure 5: Transient linear growth at the EP. $h_\text{G}$ is calculated with the parameter $\mu=1/6$ (black solid), which is the EP parameter. The red dashed line, blue dash-dotted line, and green dotted line are the amplitude factors, $h_{\text{L}}$ in Eq. \ref{['hLa=0']} with $n_{\text{L,max}}=1$, $5$, and $9$, respectively.