Pseudospectrum of horizonless compact objects: A bootstrap instability mechanism

Abstract

Recent investigations of the pseudospectrum in black hole spacetimes have shown that quasinormal mode frequencies suffer from spectral instabilities. This phenomenon may severely affect gravitational-wave spectroscopy and limit precision tests of general relativity. We extend the pseudospectrum analysis to horizonless exotic compact objects which possess a reflective surface arbitrarily close to the Schwarzschild radius, and find that their quasinormal modes also suffer from an overall spectral instability. Even though all the modes themselves decay monotonically, the pseudospectrum contours of equal resonance magnitude around the fundamental mode and the lowest overtones can cross the real axis into the unstable regime of the complex plane, unveiling the existence of non-modal pseudo-resonances. A pseudospectrum analysis further predicts that fluctuations to the system may destabilize the object when next to leading-order effects are considered, as the triggering of pseudo-resonant growth can break the order-expansion of black-hole perturbation theory.

Pseudospectrum and binary black hole merger transients

“…10 Interestingly, at intermediate timescales the dominant ringdown signal of both non-perturbed and perturbed BHs seems to be consistent with the fundamental QNM of the corresponding non-perturbed BH, even in cases in which the fundamental QNM is actually perturbed [8,14,55]. This suggests an underlying relation between the universal (asymptotic) features of BH -pseudospectra at intermediate 's and non-perturbed BH QNM spectra.…”

“…The non-perturbed and perturbed QNM problems in equation ( 40) assume the stationarity of the operator L and the perturbation δL. The same holds for the problems studied in [28,33,52,53] and, more generally, the stationarity hypothesis pervades the standard approach to the very definition of QNMs 14 . Whereas the stationarity of L is a reasonable hypothesis due to the uniqueness BH theorems, the stationarity of the perturbation δL is a strong physical hypothesis.…”

“…At the same time, this very result supports the validity of the 'close limit' in the dynamics starting at the peak of the merger. Even if we go to second-order perturbation theory, the 'close limit' approximation does not provide a good account of the transient peak, as shown in the appendix A, where the pseudospectrum is used to assess the possibility of understanding the merger peak in terms of a so-called 'pseudo-resonance', but without success (the 'pseudo-resonance' concept can be however useful in other (ultra-)compact object settings, as explored in [14]). The bottomline of this 'close-limit' approximation analysis is that a refined model is needed to explore the effective linear dynamics hypothesis, the key point being-as indicated in section 1-the correct identification of the relevant background dynamics.…”

“…Since this turns out not to be case (cf analysis in [52]) we can conclude that the second-order close-limit approximation does neither account for the merger peak: the close-limit approximation may apply from the merger, but not before 17 . This resonant mechanism may, on the contrary, prove to be of interest in the dynamical instability of ultracompact objects [14].…”

Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics

“…The hyperboloidal framework also allows us to formally define quasi-normal modes as eigenvalues of the generator of time translations for a null foliation, acting on an appropriate Hilbert space [60,61]. Furthermore, the notion of quasi-normal mode pseudospectra [21,[62][63][64][65][66] sheds light on the phenomenon of quasi-normal modes spectral instability [67][68][69], which might impact predictions from the black-hole spectroscopy programme [62,[70][71][72][73]. In the context of wave form modelling for EMRIs, the hyperboloidal approach enhanced the predictions arising from the effective-one-body approach [74][75][76][77][78][79][80][81][82], and the framework has also excelled first tests in the gravitational self-force programme [83][84][85][86][87][88].…”