IMRPhenomD composed polarizations

Scope and status

Version: imrphenomd-composed-polarizations/v1. The scalar imrphenomd_waveform.py composes the unchanged amplitude and phase specifications with the audited unit conversions. It imports no production waveform code or coefficients. Its default fits retain only the digits printed in Khan et al., arXiv:1508.07253v2, Appendix C, Table 5. Tests pass the frontend's inherited extra digits as explicit inputs; agreement conditional on those rows does not prove their publication provenance. The dimensionless QNM pair \((f_{\rm RD},f_{\rm damp})\) is also an explicit input, so this slice does not certify the remnant or QNM pipeline.

Inputs, conventions, and options

For positive physical frequency \(f\) in hertz, component masses in nominal solar masses, and luminosity distance \(d\) in megaparsecs, define \(M_s=(m_1+m_2)T_\odot^{\rm N}\), \(D_s=dL_{\rm Mpc}/c\), and \(u=M_sf\).

Mathematics and components

The composed amplitude is

\[ \mathcal A(f)=\sqrt{\frac{5\eta}{24}}\, \frac{M_s^{5/6}}{\pi^{2/3}D_s f^{7/6}}\,a(u), \]

where \(a(u)\) is the dimensionless, piecewise amplitude from the amplitude specification. The physical stationary-phase scale and inclination factors follow Cutler and Flanagan, arXiv:gr-qc/9402014v1, Eqs. (62)–(63). Khan et al. Eqs. (17)–(18) define the normalization used when fitting \(a(u)\) in their own dimensionless convention; the physical scale above is the explicit DiffGW adapter. The nominal solar mass, megaparsec and speed of light definitions are traced in the unit specification.

Let \(\phi(u)\) be the joined intrinsic phase, and \(s_{\rm peak}\) the frontend time-gauge slope using the raw fitted \(\alpha_1\) at the amplitude peak. DiffGW's phase conventions are

\[ \begin{aligned} \Psi_{\rm coal}(f)&=-\pi/4+\phi(u)-s_{\rm peak}u +2\pi f t_c-2\phi_c,\\ \Psi_{\rm ref}(f)&=\phi(u)-\phi(u_r)-s_{\rm peak}(u-u_r) +2\pi f t_c-2\phi_c,\quad u_r=M_s f_{\rm ref}. \end{aligned} \]

The reference-frequency subtraction, phase offsets, and time gauge are project conventions; the paper defines the intrinsic amplitude and phase, not these API choices. With inclination \(\iota\) in radians, the returned complex positive-frequency polarizations are

\[ \tilde h_+=\frac{1+\cos^2\iota}{2}\mathcal A e^{-i\Psi},\qquad \tilde h_\times=-i\cos\iota\,\mathcal A e^{-i\Psi}. \]

The negative exponent follows Khan et al. Eq. (7). The signed cross quadrature and choice of orbital-phase origin fix DiffGW's polarization convention. They must be kept fixed when comparing complex strains. The scalar reference returns zero strain above the inherited \(u=0.2\) default cutoff; this cutoff is a frontend choice, not a fitted paper coefficient. The present reference requires positive frequencies and covers the untapered default waveform. Additional Torch-only tapers are outside its scope.

Smooth-branch derivatives

The reference derives complex-polarization slopes from the same sourced amplitude and phase, with fit rows and the dimensionless QNM pair held fixed. Write \(H=\mathcal A e^{-i\Psi}\), let \(a'=da/du\) and \(\phi'=d\phi/du\) on the selected branch, and hold \(f_{\rm ref}\) fixed when present. The product rule gives

\[ \frac{\partial H}{\partial f}=\mathcal A e^{-i\Psi}\left[ -\frac{7}{6f}+M_s\frac{a'}{a} -i\left(M_s(\phi'-s_{\rm peak})+2\pi t_c\right)\right]. \]

The executable expression avoids division by \(a\). The polarization factors multiply this derivative unchanged. For a common rescaling \((m_1,m_2)\mapsto(\lambda m_1,\lambda m_2)\) at \(\lambda=1\), fixed spins, physical frequencies and distance, define

\[ P=u(\phi'(u)-s_{\rm peak})- \begin{cases} 0,&\text{coalescence convention},\\ u_r(\phi'(u_r)-s_{\rm peak}),&\text{reference-frequency convention}. \end{cases} \qquad \frac{\partial H}{\partial\ln\lambda} =\mathcal A e^{-i\Psi}\left(\frac56+u\frac{a'}{a}-iP\right). \]

The amplitude exponent follows the physical stationary-phase normalization; the rest follows \(u=M_sf\) and the phase adapter above. Analytic extrinsic identities are \(\partial_d\tilde h=-\tilde h/d\), \(\partial_{t_c}\tilde h=-2\pi if\tilde h\) and \(\partial_{\phi_c}\tilde h=2i\tilde h\). Inclination derivatives follow the two explicit polarization factors. The tests compare real and imaginary Torch/JAX autodiff Jacobians with these scalar expressions at interior points of all three frequency regions. Frequency and mass-scale derivatives at the hard cutoff are undefined; the tests avoid all branch boundaries. The fixed QNM pair excludes remnant and QNM response to changing mass ratio or spins from the mass-scaling check.

Implementation mapping

Component traceability and checks

Component ID	Evidence	Reference function	Torch/JAX check
`phenomd.waveform.physical_scale.v1`	Cutler–Flanagan Eqs. (62)–(63); unit definitions	`physical_scale`	Direct absolute complex strain, without amplitude rescaling
`phenomd.waveform.phase_adapter.v1`	Khan Eq. (7); DiffGW phase conventions	`phase_adapter`	Both phase conventions, nonzero \(t_c\) and \(\phi_c\)
`phenomd.waveform.polarizations.v1`	Khan Eqs. (5)–(7); Cutler–Flanagan Eq. (62); project cross sign	`polarizations`	Both complex polarizations and inclination identities
`phenomd.waveform.default_cutoff.v1`	DiffGW frontend choice	`default_cutoff`	Zero strain beyond \(u=0.2\)
`phenomd.waveform.polarization_derivatives.v1`	Product/chain rules applied to the sourced expressions and project conventions	`polarization_derivatives`	Frequency, common-mass and extrinsic complex Jacobians against both frontends

COMPONENT_BINDINGS resolves every ID to the scalar function, Torch and JAX entry points, and a maintained test.

Evidence and limitations

The FP64 cases include equal and unequal masses, aligned spins, both sides of the amplitude and phase joins, a supplied QNM pair, merger-ringdown, and a point beyond cutoff. Comparisons use direct complex error and unwrapped phase without optimizing time or phase. The test also checks the edge-on/face-on polarization identities and inverse-distance scaling analytically. The selected cases are implementation verification, not a uniform error bound or independent physical validation. A separate intrinsic composition now checks selected mass-ratio and spin derivatives with the remnant and QNM chain included. FP32, full-domain intrinsic derivatives, batch/JIT transforms, QNM endpoints and Torch's opt-in tapers remain further qualification work.

Reproduce the checks

Run python -m pytest -q tests/test_imrphenomd_waveform_math_reference.py for this slice. The full strict mathematical-reference gate also collects it.