Parametrized ringdown spin expansion coefficients: A data-analysis framework for black-hole spectroscopy with multiple events

Abstract

Black-hole spectroscopy is arguably the most promising tool to test gravity in extreme regimes and to probe the ultimate nature of black holes with unparalleled precision. These tests are currently limited by the lack of a ringdown parametrization that is both robust and accurate. We develop an observable-based parametrization of the ringdown of spinning black holes beyond general relativity, which we dub ParSpec (Parametrized Ringdown Spin Expansion Coefficients). This approach is perturbative in the spin, but it can be made arbitrarily precise (at least in principle) through a high-order expansion. It requires O(10) ringdown detections, which should be routinely available with the planned space mission LISA and with third-generation ground-based detectors. We provide a preliminary analysis of the projected bounds on parametrized ringdown parameters with LISA and with the Einstein Telescope, and discuss extensions of our model that can be straightforwardly included in the future. arXiv:1910.12893v1 [gr-qc] 28 Oct 2019where J = 1, 2, ..., q labels the mode; M i and χ i 1 are the detector-frame mass and spin of the i-th source, both measured assuming GR (see below); D is the order of the spin expansion; w (n) J and t (n) J are the dimensionless coefficients of the spin expansion for a Kerr BH in GR (provided in Table I for a few representative modes); γ i are dimensionless coupling constants, which can depend on the source i -see Eq. (6) below -but do not depend 1 The QNMs are identified by three integer numbers: the angular momentum number l, the azimuthal number m ∈ [−l, l], and the overtone number, which we set to zero in this paper, i.e. we only consider fundamental modes. For ease of notation we leave these indices implicit, i.e. ω (J) ≡ ω (0lm) , where J is an index that labels the mode. (n) J and δt (n) J (n) J and δt (n) J (or, equivalently, δW (n) J and δT (n) J ), these corrections depend only on the theory and not on the source.It is easy to check that, to leading order in α, the redefinitionsbring Eqs. (A1) and (A2) to the form in Eqs.(3) and (4) used in the main text. The above redefinitions also show that there is some degeneracy among the beyond-GR parameters, and that one can only constrain the combinations δw (n) J and δt (n) J , which contains the intrinsic mode corrections (δW (n) J and δT (n) J ) and the mass and spin corrections (δm and δχ).(n) J and δt (n) J are unconstrained by the data.

Ringing of rotating black holes in higher-derivative gravity

“…As an example, we will more specifically restrict ourselves to the range [5, 6.5]M but note that the results for the dimensionless ratios would not significantly vary when taking instead, say, [15, 19.5]M , as long as one appropriately adjusts M ref . Using the positive boundaries of the 90% confidence intervals for the (N = 160) posterior, we have made a comparison with the bounds obtained in [18]. Although there is no reason for them to match exactly, and indeed the comparison is made difficult due to the mass dependence, they are in qualitative agreement.…”

“…To determine the final black hole mass and spin we use semi-analytic formulae based on NR as determined in [61]. The main motivation to make these choices is to easily compare against the framework presented in [18]. However, we shall see that in this mass range only the lightest few events contribute significantly to the constraints, as is expected by the M 4 suppression.…”

“…However, one should be open to the possibility that there is qualitative difference between these black hole populations, not only because the huge scale difference but also because many of the LISA mergers will occur at high redshift [64]. We consider only binary black hole coalescence between black holes of mass M = 10 6 M , the lower bound on the range of masses used in [18], again with Here, M ref = 10 6 M . Note that since the reference mass is much larger than for ET, the constraints on the couplings are much weaker.…”

“…The final state as well as the QNM frequencies are sensitive to the underlying theory of gravity, and both are known to be highly constrained in GR. Hence this ringdown phase offers great potential to probe GR, and to constrain corrections to it [5,6,18].…”

Eikonal quasinormal modes of black holes beyond general relativity. II. Generalized scalar-tensor perturbations

“…A violation of this hypothesis would hint towards new physics beyond GR [17]. This has motivated considerable effort in modeling the QNM spectrum of black holes outside the Kerr paradigm, ranging from model-independent parametrisations [18][19][20][21][22][23] to calculations done in specific beyond-GR theories, e.g. [24][25][26][27][28][29][30][31].…”