High precision ringdown modeling: Multimode fits and BMS frames

Abstract

Quasinormal mode (QNM) modeling is an invaluable tool for characterizing remnant black holes, studying strong gravity, and testing general relativity. Only recently have QNM studies begun to focus on multimode fitting to numerical relativity strain waveforms. As gravitational wave observatories become even more sensitive they will be able to resolve higher-order modes. Consequently, multimode QNM fits will be critically important, and in turn require a more thorough treatment of the asymptotic frame at I þ . The first main result of this work is a method for systematically fitting a QNM model containing many modes to a numerical waveform produced using Cauchy-characteristic extraction (CCE), a waveform extraction technique which is known to resolve memory effects. We choose the modes to model based on their power contribution to the residual between numerical and model waveforms. We show that the allmode strain mismatch improves by a factor of ∼10 5 when using multimode fitting as opposed to only fitting the ð2; AE2; nÞ modes. Our most significant result addresses a critical point that has been overlooked in the QNM literature: the importance of matching the Bondi-van der Burg-Metzner-Sachs (BMS) frame of the numerical waveform to that of the QNM model. We show that by mapping the numerical waveformswhich exhibit the memory effect-to a BMS frame known as the super rest frame, there is an improvement of ∼10 5 in the all-mode strain mismatch compared to using a strain waveform whose BMS frame is not fixed. Furthermore, we find that by mapping CCE waveforms to the super rest frame, we can obtain allmode mismatches that are, on average, a factor of ∼4 better than using the publicly available extrapolated waveforms. We illustrate the effectiveness of these modeling enhancements by applying them to families of waveforms produced by numerical relativity and comparing our results to previous QNM studies.

Searching for a Ringdown Overtone in GW150914

“…We emphasize that this is achieved here by using the posterior on the time of peak strain as a prior on t 0 ; this is motivated by the observations of Refs. [17][18][19][20][21][22][23][24][25][26] described above, which show that generically the ringdown can be considered to start at around this time.…”

“…In the notation of (for example) Refs. [22,23,26], this is equivalent to using the = −m = 2 mirror modes. Or, in notation of Ref.…”

“…The claimed detection of an overtone was made possible partly because, compared to earlier studies, the authors chose to use an earlier start time for the ringdown; this was motivated by contemporary numerical relativity studies [17] (see also Refs. [18][19][20][21][22][23][24][25][26]) that demonstrated that when overtones are included the ringdown can be considered to start as early as the time of peak strain amplitude.…”

Fixing the BMS frame of numerical relativity waveforms

“…Consequently, for systems like BBHs, by performing such a mapping at u → −∞ or at u → +∞, one can wholly fix the supertranslation freedom of the system so that it agrees with PN waveforms or QNM models. Moreschi calls a frame in which the ≥ 1 components of Ψ M are zero a nice section of I + ; we, like [9,41], will instead call it the superrest frame. Furthermore, since Ψ M in the superrest frame at u → ±∞ is equal to Bondi mass M B , the supertranslation α that maps to the superrest frame at a single time can be computed with relative ease via…”

“…CCE requires freely specifying the strain on the initial null hypersurface of the simulation. As in [9,12,13,41], we choose this field to match the value and the first radial derivative of h from the Cauchy data on the world tube using the ansatz…”