IMRPhenomXHM_NSBH: NSBH Waveform Model

• IMRPhenomXHM_NSBH is a frequency-domain gravitational-wave model for quasi-circular, aligned-spin NSBH systems that includes tidal corrections and higher harmonics.
• The model blends analytical post-Newtonian tidal phasing with numerical relativity calibrations to achieve improved waveform accuracy across inspiral and merger-ringdown phases.
• Its extension to precessing configurations (IMRPhenomXPHM_NSBH) enhances parameter estimation and enables low-latency searches and multimessenger forecasts.

IMRPhenomXHM_NSBH is a frequency-domain gravitational-wave model for quasi-circular, aligned-spin neutron star–black hole binaries that includes higher-order modes beyond the dominant quadrupole, incorporates tidal effects in both phasing and amplitude, and is calibrated to numerical relativity simulations of NSBH mergers. It was introduced together with SEOBNRv5HM_ROM_NRTidalv3_NSBH as the first two frequency-domain models for this source class, and it is accompanied by a precessing extension, IMRPhenomXPHM_NSBH. The model is designed for analyses of simulated and real NSBH signals, with reported improvements over earlier NSBH waveform families and results consistent with the literature on events in the GWTC-3 and GWTC-4 catalogs (Vidal et al., 3 Jun 2026).

1. Model definition and problem setting

IMRPhenomXHM_NSBH is constructed within the PhenomXHM framework for spin-aligned compact binaries, specialized to neutron star–black hole systems through the addition of tidal structure and NSBH-specific higher-mode amplitude corrections. The target systems are quasi-circular binaries with aligned spins, and the intrinsic parameter set is written as $\lambda=\{M,\eta,\chi_1,\chi_2,\Lambda_2,\dots\}$, where the total mass, symmetric mass ratio, component spins, and neutron-star tidal deformability enter explicitly (Vidal et al., 3 Jun 2026).

The model addresses a regime in which both matter effects and mode content can be observationally relevant. Earlier NSBH models such as PhenomNSBH and SEOBNRv4_ROM_NSBH are described in the reported comparisons as restricted to the $(2,2)$ mode, whereas IMRPhenomXHM_NSBH extends the waveform representation to higher harmonics and supplements the black-hole-binary baseline with tidal corrections in phase and amplitude. In the performance discussion, the model is also denoted XNSBH.

2. Frequency-domain decomposition and tidal phasing

The frequency-domain strain is decomposed into spin-$(-2)$ spherical harmonics,

$$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$

with each complex mode written in polar form,

$$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$

In the LVK LALSuite convention, only $m<0$ modes are modeled directly, while $m>0$ modes follow from

$h_{\ell m}(f)=(-1)^\ell h^*_{(\ell,-m)}(-f).$

This modewise formulation makes the higher-mode extension explicit and provides the interface through which tidal structure is incorporated (Vidal et al., 3 Jun 2026).

The total phase of each mode is decomposed as

$$\psi_{\ell m}(f)=\psi_{\ell m}^{\rm BBH}(f)+\psi_{\ell m}^{T}(f).$$

For the $(2,2)$ mode, the tidal phase is supplied by NRTidalv3 in a closed-form “Rational” PN-resummed form,

$(2,2)$0

where $(2,2)$1, $(2,2)$2 is the dynamical tidal parameter proportional to $(2,2)$3, and $(2,2)$4 is a $(2,2)$5-order Padé approximant that tends to unity at low $(2,2)$6 so as to recover the 7.5PN expansion. Higher-mode tidal phases are introduced through the mode scaling

$(2,2)$7

Spin-induced quadrupole and octupole tidal terms up to 3.5PN are also added as in Dietrich et al. (Vidal et al., 3 Jun 2026).

3. Amplitude construction and numerical-relativity calibration

The NSBH amplitude is built multiplicatively on the underlying BBH PhenomXHM amplitude, but the construction is piecewise in frequency. In the inspiral region, for $(2,2)$8,

$(2,2)$9

Here $(-2)$0 comes from the stationary-phase approximation of the known PN tidal-amplitude terms up to 2.5PN in Henry et al. and Dones et al., with Love numbers and spin-quadrupole coefficients fixed by universal I-Love-Q fits (Vidal et al., 3 Jun 2026).

In the merger-ringdown region, for $(-2)$1,

$(-2)$2

with suppression function

$(-2)$3

where $(-2)$4 are determined mode-by-mode. The fit is performed at collocation frequencies defined by

$(-2)$5

$(-2)$6

and

$(-2)$7

At each collocation point, the ratio

$(-2)$8

is fitted in $(-2)$9, $$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$0, and $$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$1 to the ansatz

$$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$2

with the full eight-parameter form given in App. A.2 of the paper (Vidal et al., 3 Jun 2026).

Between inspiral and merger-ringdown, a smooth one-parameter window $$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$3 connects the two amplitude pieces between $$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$4 and $$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$5. The window is built from

$$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$6

This construction provides a continuous interpolation between the PN-corrected inspiral amplitude and the NR-calibrated suppression of the high-frequency BBH baseline.

4. Spin treatment and the precessing extension

Aligned-spin effects enter through the PhenomXHM baseline’s use of the effective spin,

$$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$7

IMRPhenomXHM_NSBH is specifically the spin-aligned model for NSBH binaries. Its scope therefore excludes generic precession at the base level, even though tidal spin effects are included in the phase sector (Vidal et al., 3 Jun 2026).

The associated precessing extension is IMRPhenomXPHM_NSBH, also referred to in the summary as XPNSBH. It “twists up” the aligned-spin NSBH waveform in a co-precessing frame exactly as in IMRPhenomXPHM for BBH. The time-dependent Euler angles are obtained either from the closed-form MSA solution of PN spin precession due to Chatziioannou et al. or by integrating SpinTaylorT4. This establishes a direct relation between the aligned-spin NSBH construction and the broader PhenomXPHM strategy for precessing systems.

5. Quantitative performance and parameter estimation

Against hybrid NR waveforms built from NRSurTidal and NR, the reported mismatches for four aligned-spin NSBH hybrids with mass ratios $$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$8 are at median levels $$h(f,\theta,\phi;\lambda)=\sum_{\ell\ge 2}\sum_{m=-\ell}^{\ell} {}_{-2}Y_{\ell m}(\theta,\phi)\, h_{\ell m}(f;\lambda),$$9 over 25–35 extrinsic orientations for XNSBH and v5HMROM_NSBH. By contrast, the earlier $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$0-only NSBH models PhenomNSBH and SEOBNRv4_ROM_NSBH lie at $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$1. The inclusion of higher modes reduces the spread in mismatch across inclinations by a factor $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$2 (Vidal et al., 3 Jun 2026).

The parameter-estimation injections reported for hybrid signals SXS:0001 and SXS:0002 at network $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$3 illustrate two distinct regimes. For SXS:0001, a non-disruptive system with $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$4, PhenomNSBH recovers

$$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$5

whereas XNSBH gives

$$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$6

The Bayes factors strongly favor higher modes, with $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$7. For SXS:0002, a disruptive system with $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$8, only the NSBH models yield $$h_{\ell m}(f)\equiv A_{\ell m}(f)e^{-i\psi_{\ell m}(f)}.$$9 posteriors peaking near 791, and v5HMROM_NSBH recovers

$m<0$0

For real-event reanalyses of GW200105, GW200115, GW230518, and GW230529, all models give broadly consistent posteriors at $m<0$1, although small shifts occur in chirp mass, distance, and $m<0$2 when tides or higher modes are added. A specific example is GW230518, for which XNSBH shifts the luminosity distance from

$m<0$3

with PhenomNSBH to

$m<0$4

The abstract summarizes these studies by stating that the new models show clear improvements over their predecessors in analyses of simulated signals while yielding results consistent with the literature when applied to real events from the GWTC-3 and GWTC-4 catalogs (Vidal et al., 3 Jun 2026).

6. Domain of validity, calibration limits, and usage

The stated domain of validity is quasi-circular NSBH with aligned spins up to $m<0$5 and $m<0$6. The practical mass-ratio range is $m<0$7 from 1 to approximately 20, with neutron-star mass $m<0$8 and $m<0$9. Tidal spin effects are included, but the neutron-star spin is set to zero in the amplitude calibration (Vidal et al., 3 Jun 2026).

The amplitude fits are calibrated to NR datasets with $m>0$0, $m>0$1, and piecewise-polytropic equations of state; extrapolation beyond this region requires caution. An additional limitation arises from the phasing sector: the NRTidalv3 tidal phase is calibrated on binary neutron stars up to $m>0$2, and its high-frequency rational form is “capped” by a second-order Taylor extension beyond its first pole or inflection. These caveats delimit the regime in which the waveform can be treated as directly calibrated rather than extrapolative.

The summary characterizes the model as combining a highly efficient frequency-domain phenomenological framework with state-of-the-art tidal phasing and calibrated higher-mode amplitudes, with a speed of approximately 1 s per waveform on a modern CPU. The listed applications are low-latency searches, rapid parameter estimation in LVK O4/O5 and beyond, population studies of NSBH equation-of-state effects, and multimessenger forecasts of tidal-disruption or kilonova signatures in next-generation detectors. A plausible implication is that the model is intended not only as a waveform-accuracy update over single-mode NSBH approximants, but also as an analysis-ready representation for computationally intensive inference pipelines.