Considerations on the excitation of black hole quasinormal modes

We provide some considerations on the excitation of black hole quasinormal modes (QNMs) in different physical scenarios. Considering a simple model in which a stream of particles accretes onto a black hole, we show that resonant QNM excitation by hyperaccretion requires a significant amount of fine-tuning, and is quite unlikely to occur in nature. Then we summarize and discuss present estimates of black hole QNM excitation from gravitational collapse, distorted black holes and head-on black hole collisions. We emphasize the areas that, in our opinion, are in urgent need of further investigation from the point of view of gravitational wave source modeling. A leading candidate source of detectable waves for Earth-based interferometers such as LIGO, Virgo, GEO600 and TAMA, as well as for the space-based interferometer LISA, is the inspiral and merger of binary black holes. The waveform should comprise three parts, usually referred to as inspiral, merger and ringdown. The inspiral waveform, originating from that part of the decaying orbit leading up to the innermost stable orbit, can be analyzed using post-Newtonian theory. Extensive studies of the detectability of this phase of the signal have been carried out, both for Earth-based [1] and for spacebased interferometers [2]. The nature of the merger waveform is largely unknown at present, and is the subject of work in numerical relativity. The ringdown waveform originates from the distorted final black hole, and consists of a superposition of quasi-normal modes (QNMs). Each mode has an oscillation frequency and a damping time that are uniquely determined by the mass M and specific angular momentum j ≡ J/M2 of the black hole [3]. The amplitudes and phases of the various modes are determined by the specific process that formed the final hole. The uniqueness of the modes’ frequencies and damping times is directly related to the “no hair” theorem of general relativistic black holes. Therefore a reliable detection and accurate identification of QNMs could provide the “smoking gun” for black holes and an important test of general relativity in the strong-field regime [4]. The detectability of ringdown waves, and the accuracy with which we can measure their frequencies and damping times to test the no-hair theorem, depend mainly on the energy carried by each mode [5]. In turn, the energy distribution depends on the details of the merger process. Given our poor understanding of the merger phase, we have at best sketchy information concerning the energy distribution between different modes. In the first part of this paper we show by a simple toy model that QNM excitation by infalling matter requires a significant amount of fine-tuning, and is quite unlikely in astrophysically realistic scenarios. In the second part we briefly review present estimates of QNM excitation in various physical situations, including gravitational collapse, simulations of single distorted black holes and head-on collisions of two black holes. This second part is an extended version of Sec. VB in [5]. Most of our considerations can be applied to the solar-mass black holes detectable by Earth-based interferometers, but the main motivation for this short review comes from our study of the massive black holes detectable by LISA [5]. Throughout this paper we use geometrical units (G = c = 1). EXCITATION BY INFALLING MATTER Various authors [6, 7] suggested that quasinormal ringing could be resonantly excited by clumps of matter falling into the black hole at some appropriate rate. This rate should be such that the spatial separation of the clumps equals a multiple of the typical QNM wavelength. Here we show by a simple model that, in principle, the modes of a black hole can indeed be excited in this way. We also show that a simple addition of damped sinusoids provides a good fit of the resulting gravitational waveforms: in other words, we can interpret the resonant excitation of the modes in terms of interference between gravitational waves. For simplicity we consider clumps of mass μ much smaller than the black hole mass, μ ≪ M, so that we can apply a perturbative analysis. Consider first one single clump falling from infinity. This process was first analyzed in a classical paper by Davis, Ruffini, Press and Price [8] (hereafter DRPP). They found that the total radiated energy is ≃ 0.0104μ2/M, most of it (0.0091μ2/M) being emitted as quadrupolar (l= 2) waves, and that the total energy spectrum is peaked at ω ∼ 0.32/M, very close to the fundamental l = 2 QNM frequency ω ≃ 0.3737/M. We can now superpose the waveform and its Fourier transform for the one-particle case to represent two or more particles falling into a black hole and to study interference phenomena (we are assuming that the clumps are non-interacting, so that there is no extra contribution to the total energy-momentum tensor). A similar sum-over-point-particles approach has been used many times in the past [9], but the particular process we consider here has not been studied before. Let us first suppose that we drop two particles with a temporal separation of T (which, bearing in mind that the retarded time is the measured quantity, also means a spatial separation of T between the two bodies). If Ψ1(t,r) represents the waveform for the first body (normalized by μ , where μ is the mass of the infalling body), then the total waveform will be Ψ(t,r) = μΨ1(t,r)+ μΨ1(t −T,r) . (1) Likewise the Fourier transform Ψ̃(ω,r) will be Ψ̃ = μΨ̃1 + μeiωT Ψ̃1 . (2) For N bodies dropped regularly, such that the temporal difference between the dropping of the first and of the last is T , we have Ψ(t,r) = N−1 ∑ j=0 μΨ1 (