TEOBResumSPA_NRPMw Waveform Model

• TEOBResumSPA_NRPMw waveform is a comprehensive model that combines TEOBResumSPA for inspiral–merger phases with NRPMw for the kilohertz postmerger, capturing the full gravitational-wave signal.
• It uses analytically derived frequency-domain expressions with complex Gaussian wavelets to model the modulated amplitude and frequency features of postmerger signals.
• Calibrated against hundreds of numerical-relativity simulations, the model links binary parameters via EOS-insensitive relations to accurately predict postmerger peak frequencies and waveform morphology.

Searching arXiv for TEOBResumSPA, NRPMw, and closely related TEOBResumS papers to ground the article in the relevant literature. The TEOBResumSPA_NRPMw waveform is a complete inspiral–merger–postmerger waveform model for binary neutron star systems that combines TEOBResumSPA—or its time-domain parent TEOBResumD—for the inspiral and merger up to the time-domain peak of the $(2,2)$ amplitude, with NRPMw, a numerical-relativity–informed frequency-domain model of the postmerger kilohertz signal from the remnant (Breschi et al., 2022). In this construction, the inspiral–merger sector inherits the effective-one-body dynamics, resummed multipolar amplitudes, NR-informed next-to-quasi-circular structure, and acceleration strategies developed in the TEOBResumS family (Nagar et al., 2018, Riemenschneider et al., 2021), while the postmerger sector is modeled analytically as a sum of complex Gaussian wavelets calibrated to numerical-relativity simulations through EOS-insensitive relations (Breschi et al., 2022). The result is a single waveform family intended to cover the signal from low frequencies through merger and postmerger out to several kilohertz (Breschi et al., 2022).

1. Definition and signal domain

TEOBResumSPA_NRPMw targets the gravitational-wave signal of binary neutron star remnants after the amplitude peak of merger, while retaining a consistent inspiral–merger description from the TEOBResumS/TEOBResumSPA framework (Breschi et al., 2022). The postmerger regime modeled by NRPMw is the dominant $(\ell,m)=(2,2)$ multipole emitted from the time when the two neutron-star cores have fused and a massive, hot, differentially rotating remnant has formed, up to the point where the remnant collapses to a black hole or relaxes to a long-lived neutron star on secular timescales (Breschi et al., 2022).

In this regime, the signal is dominated by quasi-periodic oscillations at a nearly constant frequency $f_2 \sim 2\!-\!4$ kHz, shows broad spectral peaks rather than discrete lines, with full-width-at-half-maximum typically $300{-}600$ Hz, exhibits strong amplitude and frequency modulations associated with quasi-radial oscillations of the remnant at a characteristic frequency $f_0 \sim \mathcal{O}(1\,{\rm kHz})$, and may end abruptly when the remnant collapses to a black hole (Breschi et al., 2022). TEOBResumSPA provides $\tilde h_{\rm insp}(f)$ up to the merger frequency $f_{\rm mrg}$, whereas NRPMw supplies $\tilde h_{\rm PM}(f)$ in the kilohertz range, typically from $\sim 1$ kHz up to $\sim 8$ kHz, the band used in NR faithfulness tests with ET-D noise (Breschi et al., 2022).

The inspiral–merger component is rooted in the TEOBResumS family of effective-one-body models. In that framework, the waveform is constructed mode by mode from resummed EOB dynamics, a multipolar flux, NR-informed NQC corrections, and an NR-informed merger–ringdown attachment (Riemenschneider et al., 2021). The TEOBResumSPA variant is the stationary-phase–approximant representation of the same underlying EOB dynamics, and the post-adiabatic approximation provides an efficient route to compute inspiral waveforms up to a few orbits before merger with phase differences below $(\ell,m)=(2,2)$0 rad in representative BBH and BNS cases (Nagar et al., 2018).

2. Frequency-domain postmerger construction

NRPMw models the postmerger waveform as a sum of complex Gaussian wavelets with amplitude and frequency modulations (Breschi et al., 2022). Its basic time-domain building block is

$(\ell,m)=(2,2)$1

with complex coefficients $(\ell,m)=(2,2)$2 and support $(\ell,m)=(2,2)$3 (Breschi et al., 2022). The Fourier transform can be written analytically as

$(\ell,m)=(2,2)$4

where

$(\ell,m)=(2,2)$5

and a global time shift is implemented through

$(\ell,m)=(2,2)$6

For $(\ell,m)=(2,2)$7, the wavelet reduces to a damped sinusoid and $(\ell,m)=(2,2)$8 becomes a Lorentzian (Breschi et al., 2022).

Amplitude modulations are introduced through

$(\ell,m)=(2,2)$9

which in frequency domain yields sidebands at $f_2 \sim 2\!-\!4$0, corresponding to the secondary spectral peaks $f_2 \sim 2\!-\!4$1 (Breschi et al., 2022). Frequency modulations act on the instantaneous angular frequency,

$f_2 \sim 2\!-\!4$2

through

$f_2 \sim 2\!-\!4$3

with $f_2 \sim 2\!-\!4$4, yielding an analytic frequency-domain representation with multiple sidebands at $f_2 \sim 2\!-\!4$5 (Breschi et al., 2022).

The dominant $f_2 \sim 2\!-\!4$6 postmerger mode is represented as

$f_2 \sim 2\!-\!4$7

where the four components are physically motivated (Breschi et al., 2022). $f_2 \sim 2\!-\!4$8 models the early postmerger fusion stage from merger time $f_2 \sim 2\!-\!4$9 to the first amplitude minimum at $300{-}600$0; $300{-}600$1 models the first bounce of the merged core from $300{-}600$2 to the first postmerger amplitude maximum $300{-}600$3; $300{-}600$4 describes the strongly dynamical pulsating phase from $300{-}600$5 to $300{-}600$6 with both AM and FM active; and $300{-}600$7 models the late damped tail from $300{-}600$8 to the collapse time $300{-}600$9 or to a large cutoff for long-lived remnants (Breschi et al., 2022). A fifth piece $f_0 \sim \mathcal{O}(1\,{\rm kHz})$0 could represent black-hole ringdown after collapse, but it is set to zero in this work because its power is negligible in the 1–8 kHz band for current/ET sensitivities (Breschi et al., 2022).

The corresponding frequency-domain waveform is

$f_0 \sim \mathcal{O}(1\,{\rm kHz})$1

with each term an analytical function of the wavelet and modulation parameters (Breschi et al., 2022). The dominant peak at $f_0 \sim \mathcal{O}(1\,{\rm kHz})$2 corresponds to the main quadrupole oscillation of the remnant, while sidebands at $f_0 \sim \mathcal{O}(1\,{\rm kHz})$3 and $f_0 \sim \mathcal{O}(1\,{\rm kHz})$4 arise from AM/FM coupling with the quasi-radial mode at $f_0 \sim \mathcal{O}(1\,{\rm kHz})$5 (Breschi et al., 2022).

3. Calibration dataset and EOS-insensitive mapping

NRPMw is calibrated on 618 numerical-relativity simulations spanning total mass $f_0 \sim \mathcal{O}(1\,{\rm kHz})$6, mass ratio $f_0 \sim \mathcal{O}(1\,{\rm kHz})$7, symmetric tidal coupling parameter $f_0 \sim \mathcal{O}(1\,{\rm kHz})$8, aligned spins $f_0 \sim \mathcal{O}(1\,{\rm kHz})$9, and 21 EOSs, including 7 finite-temperature microphysical EOSs and 14 piecewise-polytropic EOSs (Breschi et al., 2022). Some EOSs include quark deconfinement or hyperons, and the dataset is constructed from three independent NR codes, {\sc BAM}, {\sc THC}, and the Kyoto/Tokyo code (Breschi et al., 2022).

The binary parameter vector used to map inspiral information into postmerger quantities is

$\tilde h_{\rm insp}(f)$0

with tidal coupling constant

$\tilde h_{\rm insp}(f)$1

(Breschi et al., 2022). For any postmerger quantity $\tilde h_{\rm insp}(f)$2, the fit ansatz is

$\tilde h_{\rm insp}(f)$3

where

$\tilde h_{\rm insp}(f)$4

and the tidal factor is the rational function

$\tilde h_{\rm insp}(f)$5

(Breschi et al., 2022).

The postmerger parameter vector is

$\tilde h_{\rm insp}(f)$6

and the subset mapped from inspiral parameters is

$\tilde h_{\rm insp}(f)$7

(Breschi et al., 2022). The remaining parameters $\tilde h_{\rm insp}(f)$8 are treated as free or fitted directly to data (Breschi et al., 2022).

Several relations are comparatively robust. The merger amplitude $\tilde h_{\rm insp}(f)$9 has a $f_{\rm mrg}$0 error of about 1.8%, the merger frequency $f_{\rm mrg}$1 about 2.6%, and the main postmerger peak $f_{\rm mrg}$2 about 3.9%, corresponding to roughly 100 Hz near 2.5 kHz (Breschi et al., 2022). By contrast, $f_{\rm mrg}$3, the postmerger nodal amplitudes, and the FM parameters $f_{\rm mrg}$4 are substantially less constrained, with the FM parameters exhibiting errors of $f_{\rm mrg}$5 (Breschi et al., 2022). EOSs with strong phase transitions can mildly violate the $f_{\rm mrg}$6 relation at the $f_{\rm mrg}$7 level in special cases, though the relation remains overall robust (Breschi et al., 2022).

4. Coupling to TEOBResumSPA and EOB foundations

The inspiral–merger sector attached to NRPMw inherits the effective-one-body dynamics and waveform factorization of the TEOBResumS family (Nagar et al., 2018, Riemenschneider et al., 2021). In the nonspinning multipolar construction, the EOB Hamiltonian is

$f_{\rm mrg}$8

with

$f_{\rm mrg}$9

(Nagar et al., 2019). The radial potential is written as a PN series augmented by an effective 5PN coefficient $\tilde h_{\rm PM}(f)$0, then Padé-resummed as

$\tilde h_{\rm PM}(f)$1

in the nonspinning multipolar model (Nagar et al., 2019). The calibrated NR-informed fit for the Padé-resummed, 6PN-hybrid amplitude model is

$\tilde h_{\rm PM}(f)$2

with numerical coefficients reported in the paper (Nagar et al., 2019).

The inspiral waveform is decomposed into spherical-harmonic multipoles,

$\tilde h_{\rm PM}(f)$3

and each orbital mode is factorized as

$\tilde h_{\rm PM}(f)$4

(Nagar et al., 2019). The improved nonspinning multipolar model upgrades the residual amplitude corrections $\tilde h_{\rm PM}(f)$5 by hybridizing full 3PN $\tilde h_{\rm PM}(f)$6-dependent information with test-mass results up to 6PN order and then Padé-resumming them (Nagar et al., 2019). The calibrated model includes NR-completed higher modes for $\tilde h_{\rm PM}(f)$7, $\tilde h_{\rm PM}(f)$8, $\tilde h_{\rm PM}(f)$9, $\sim 1$0, $\sim 1$1, $\sim 1$2, $\sim 1$3, $\sim 1$4, and $\sim 1$5, while other modes up to $\sim 1$6 remain purely analytical (Nagar et al., 2019).

The TEOBResumS family also uses NR-informed NQC corrections and can be accelerated by a post-adiabatic treatment of the inspiral. In that approximation, the EOB momenta are solved algebraically on a radial grid and the time and phase are recovered by quadratures, yielding waveforms more than 100 times faster than a standard ODE solver in a nonoptimized Matlab implementation for a standard BNS system from 10 Hz (Nagar et al., 2018). A later assessment of consistent NQC corrections and PA approximation in multipolar BBH waveforms showed EOB/NR unfaithfulness well below $\sim 1$7 over 611 NR simulations, with 78.5% of cases below $\sim 1$8 for $\sim 1$9 using Advanced LIGO noise (Riemenschneider et al., 2021). This suggests that the TEOBResumSPA inspiral attached to NRPMw is designed to inherit an NR-calibrated EOB backbone rather than a purely PN one.

5. Joining prescription, recalibration, and parameter inference

TEOBResumSPA_NRPMw is assembled by joining a frequency-domain inspiral–merger waveform $\sim 8$0 to a frequency-domain remnant waveform $\sim 8$1 near the merger frequency $\sim 8$2 (Breschi et al., 2022). In practice, one computes $\sim 8$3 and $\sim 8$4 from the NRPMw quasiuniversal relations, chooses a joining frequency $\sim 8$5 close to $\sim 8$6, fixes $\sim 8$7 and $\sim 8$8 in the fusion wavelet to match TEOBResumSPA at merger, and sets the time origin at the TEOBResumSPA merger time so that postmerger nodal times are referenced consistently (Breschi et al., 2022).

To avoid sharp spectral features, the transition is smoothed by tapering or blending: $\sim 8$9 where $(\ell,m)=(2,2)$00 is a Planck-like or raised-cosine window over a small interval around $(\ell,m)=(2,2)$01 (Breschi et al., 2022). Because the fusion wavelet is parameterized by $(\ell,m)=(2,2)$02, $(\ell,m)=(2,2)$03, and $(\ell,m)=(2,2)$04, the full waveform is phase-coherent across inspiral, merger, and postmerger (Breschi et al., 2022).

A central feature of NRPMw is its use of recalibration parameters $(\ell,m)=(2,2)$05 to represent the residual scatter of the EOS-insensitive relations (Breschi et al., 2022). For any calibrated quantity,

$(\ell,m)=(2,2)$06

with $(\ell,m)=(2,2)$07 treated as a dimensionless fractional correction (Breschi et al., 2022). In Bayesian analyses, $(\ell,m)=(2,2)$08 is assigned a Gaussian prior with mean zero and variance equal to the variance of the NR residuals. This enables marginalization over theoretical modeling errors and allows the waveform to behave as an informed but flexible postmerger model (Breschi et al., 2022). The same framework also permits tests of quasiuniversality: if data require large $(\ell,m)=(2,2)$09 inconsistent with the priors, that may indicate an EOS beyond the calibration assumptions, such as a strong phase transition (Breschi et al., 2022).

In a parameter-estimation pipeline such as the bajes code mentioned in the paper, one samples

$(\ell,m)=(2,2)$10

imposes inspiral–postmerger consistency through $(\ell,m)=(2,2)$11, chooses priors for the free postmerger parameters $(\ell,m)=(2,2)$12 and $(\ell,m)=(2,2)$13, includes $(\ell,m)=(2,2)$14 with Gaussian priors, and evaluates the likelihood over the full frequency band, for example 10 Hz–8 kHz (Breschi et al., 2022). The output constrains masses, spins, tidal parameters, postmerger peak frequencies, collapse time, and possible deviations from EOS-insensitive relations (Breschi et al., 2022).

6. Validation, scope, and limitations

NRPMw was validated on 102 independent NR simulations not used in calibration, using ET-D noise in the frequency interval $(\ell,m)=(2,2)$15 kHz (Breschi et al., 2022). The faithfulness is defined by maximizing the overlap over overall time and phase shifts,

$(\ell,m)=(2,2)$16

(Breschi et al., 2022). Without recalibration, and optimizing only over $(\ell,m)=(2,2)$17, the median unfaithfulness is $(\ell,m)=(2,2)$18, with about 38% of cases below 0.1 (Breschi et al., 2022). When all recalibration parameters are allowed to vary within $(\ell,m)=(2,2)$19 of their NR-based priors, the median unfaithfulness improves to about $(\ell,m)=(2,2)$20, about 94% of the validation sample lies below 0.1, and many long-lived and unequal-mass cases reach $(\ell,m)=(2,2)$21 (Breschi et al., 2022).

The validated regime is explicitly limited to $(\ell,m)=(2,2)$22, $(\ell,m)=(2,2)$23, $(\ell,m)=(2,2)$24, $(\ell,m)=(2,2)$25, and EOSs similar to those used in calibration (Breschi et al., 2022). Extrapolation may be unreliable for very high total masses near prompt collapse, extreme mass ratios $(\ell,m)=(2,2)$26, large spins $(\ell,m)=(2,2)$27, or EOSs with strong, sharp phase transitions not represented in the training set (Breschi et al., 2022). The FM parameters are particularly uncertain and may need to be treated as free or strongly recalibrated parameters, as they can encode effects such as turbulent viscosity (Breschi et al., 2022).

The inspiral–merger foundation also has model-dependent systematics. In the TEOBResumS multipolar EOB literature, explicit mode mixing is neglected in the ringdown treatment of modes such as $(\ell,m)=(2,2)$28, $(\ell,m)=(2,2)$29, $(\ell,m)=(2,2)$30, and $(\ell,m)=(2,2)$31, and this simplification can mildly affect unfaithfulness while keeping it below the percent-to-few-percent level in the tested BBH ranges (Nagar et al., 2019). Later analyses of analytic systematics in TEOBResumS emphasized that correct control of the noncircular part of the dynamics during late plunge is essential, and that simply adding higher-PN information without retuning can worsen EOB/NR agreement (Nagar et al., 2023). This suggests that, in TEOBResumSPA_NRPMw, the inspiral and postmerger parts should be understood as jointly calibrated components rather than separable approximations.

7. Scientific role and extensions within the TEOBResumS family

Within the TEOBResumS family, TEOBResumSPA_NRPMw occupies the role of a frequency-domain, NR-informed waveform that links a calibrated EOB inspiral to a phenomenological but physics-motivated BNS postmerger (Breschi et al., 2022, Albanesi et al., 18 Mar 2025). The broader TEOBResumS program has expanded from quasi-circular spin-aligned BBH and BNS models to generic-orbit frameworks such as TEOBResumS-Dalí, which incorporates tidal interactions, generic spins, multipolar radiation reaction and waveform, and NR information for arbitrary orbits (Albanesi et al., 18 Mar 2025). That work explicitly states that the model inherits both the post-adiabatic approximation for fast quasi-circular dynamics and the EOBSPA for the computation of frequency-domain EOB waveforms (Albanesi et al., 18 Mar 2025). It also embeds NRPM or NRPMw as the postmerger completion for BNS systems, so that the model delivers a representation of the complete BNS spectrum within a unified EOB framework (Albanesi et al., 18 Mar 2025).

For BNS inspirals, the tidal sector of TEOBResum has itself been refined through resummed gravitoelectric and gravitomagnetic tidal potentials and tidal multipolar waveform corrections (Akcay et al., 2018). That improved nonspinning tidal model validated EOB energetics and phasing against NR and identified the resummed gravitoelectric octupolar term as the most important new conservative contribution, capable of producing up to 1 rad of dephasing relative to its nonresummed version depending on the neutron-star model (Akcay et al., 2018). This provides additional context for TEOBResumSPA_NRPMw: the inspiral tidal parameters that determine $(\ell,m)=(2,2)$32 and enter the EOB Hamiltonian are also the parameters used by NRPMw to predict postmerger frequencies and morphology (Breschi et al., 2022, Akcay et al., 2018).

A common misconception is that NRPMw is merely an agnostic spectral fit. The construction described in the literature is more constrained: its wavelet parameters are tied to binary masses, tidal deformabilities, and aligned spins through EOS-insensitive relations, and the recalibration parameters are explicitly introduced to encode the residuals of those relations rather than to replace them (Breschi et al., 2022). Conversely, it would also be misleading to treat TEOBResumSPA_NRPMw as fully universal outside its calibration region. The stated parameter ranges, EOS coverage, and spin limits are integral to the model definition (Breschi et al., 2022).

Taken together, these elements define TEOBResumSPA_NRPMw as a waveform architecture in which the inspiral–merger signal is provided by a calibrated EOB stationary-phase model and the kilohertz remnant signal is supplied by an analytical frequency-domain postmerger representation informed by hundreds of numerical-relativity simulations (Breschi et al., 2022). This suggests an overview between low-frequency tidal inference and kilohertz spectroscopy: the same binary parameters that shape the inspiral phase also determine the postmerger peak frequency $(\ell,m)=(2,2)$33, sideband structure, and collapse behavior within a single coherent model (Breschi et al., 2022).