Gravitational-wave spectroscopy of massive black holes with the space interferometer LISA

Abstract

Newly formed black holes are expected to emit characteristic radiation in the form of quasi-normal modes, called ringdown waves, with discrete frequencies. LISA should be able to detect the ringdown waves emitted by oscillating supermassive black holes throughout the observable Universe. We develop a multi-mode formalism, applicable to any interferometric detectors, for detecting ringdown signals, for estimating black hole parameters from those signals, and for testing the no-hair theorem of general relativity. Focusing on LISA, we use current models of its sensitivity to compute the expected signal-to-noise ratio for ringdown events, the relative parameter estimation accuracy, and the resolvability of different modes. We also discuss the extent to which uncertainties on physical parameters, such as the black hole spin and the energy emitted in each mode, will affect our ability to do black hole spectroscopy.

Numerical study of the quasinormal mode excitation of Kerr black holes

“…In fact, as stressed (for example) in Ref. [8,29], QNMs of Kerr black holes always come "in pairs". In the Kerr case, for a given multipole (ℓ, m) we have to solve an eigenvalue problem to determine both the quasinormal frequencies ω ℓmn and the angular separation constant A ℓmn (not to be confused with the mode amplitude A ℓmn introduced below), used to separate the angular and radial dependence of the Teukolsky equation and write it as two ordinary differential equations.…”

“…6 of [8] for an illustration of this). For m = 0 (or for any value of m in the Schwarzschild case) the two "mirror solutions" are degenerate in modulus of the frequency and damping time.…”

Formation of supermassive black holes

“…If the mass of the seeds is below ∼ 10 5 M , their flux is too weak for single sources to be detected electromagnetically. Seeds of mass ∼ < 10 5 M can nevertheless be directly identified during their mergers, by detecting their emission of gravitational radiation (Hughes, 2002;Berti et al, 2006). Additionally, gravitational waves produced during the inspirals of compact objects into MBHsextreme-mass-ratio inspirals (EMRIs) are expected to provide accurate constraints on the population of MBHs in the 10 4 M -10 7 M range (Gair et al 2010), which is the mass range where we can expect 'memory' of the initial conditions, as detailed in section 4.4.…”

Sources of Gravitational Waves: Theory and Observations

“…and "44 QNM freq. ", see [282] for QNM frequencies) that are expected to carry a significant amount of energy. Although BHs have an infinite sequence of modes of higher frequencies, numerical simulations of BH mergers reveal that they are devoid of any appreciable energy in modes with > 4 [302] and so we do not expect sources to radiate significantly in the top shaded region.…”

“…Following the closelimit result [276], in a first approximation the plunge and QNM signals are matched at the light ring (i.e., at the unstable photon circular orbit), where the peak of the potential barrier around the newborn BH is located. Thus, the EOB merger-ringdown waveform is built as a linear superposition of QNMs of the final Kerr BH [66,281] h merger-RD 20) where N is the number of overtones [282,283], A mn is the complex amplitude of the n-th overtone, and σ mn = ω mn − i/τ mn is the complex frequency of this overtone with positive (real) frequency ω mn and decay time τ mn . The complex QNM frequencies are known functions of the mass and spin of the final Kerr BH.…”