IMRPhenomD intrinsic phase and joins

Scope and status

imrphenomd-phase-joins-paper-precision/v1 is the executable phase specification in imrphenomd_phase.py. Its primary source is Khan et al., Paper II, arXiv:1508.07253v2, Eqs. (13)–(16), (27)–(28), (31), (37), Appendix B and Table 5. The SymPy module imports no production waveform implementation or coefficients.

Inputs, conventions, and options

Let \(u=Mf\) be frequency in the initial total mass, \(v=(\pi u)^{1/3}\), \(\eta=m_1m_2/M^2\), \(\delta=(m_1-m_2)/M\), \(\chi_s=(\chi_1+\chi_2)/2\), and \(\chi_a=(\chi_1-\chi_2)/2\). With \(M\) in seconds, the frontend frequency in Hz is \(f=u/M\). The amplitude specification defines the same spin coordinates and supplies the amplitude peak; the remnant specification supplies the positive \(f_{\rm RD}\) and \(f_{\rm damp}\) in total-mass units.

Mathematics and components

The aligned-spin TaylorF2 phase from Paper II Eq. (27) is

\[\phi_{\rm PN}(u)=\frac{3}{128\eta v^5} \left[1+c_2v^2+c_3v^3+c_4v^4 +(c_5+c_{5\log}\log v)v^5 +(c_6+c_{6\log}\log(4v))v^6+c_7v^7\right].\]

The exact rational, \(\pi\), and Euler-constant coefficients are in pn_phase_coefficients, transcribed from Appendix B Eqs. (46)–(53). In particular, the logarithmic terms retain the paper's distinct \(\log v\) and \(\log(4v)\) bases. The arbitrary coalescence time and phase constants are excluded from this intrinsic function.

Paper II Eq. (28) adds four calibrated inspiral corrections:

\[\phi_{\rm ins}(u)=\phi_{\rm PN}(u)+\frac{1}{\eta} \left(\sigma_1u+\frac34\sigma_2u^{4/3} +\frac35\sigma_3u^{5/3}+\frac12\sigma_4u^2\right).\]

The arbitrary constant \(\sigma_0\) is set to zero; the frontend phase conventions below supply the global phase offset.

The middle and merger-ringdown phase integrals of Eqs. (16) and (14) are

\[\begin{aligned} \phi_{\rm int}(u)&=\frac1\eta\left(\beta_0+\beta_1u +\beta_2\log u-\frac{\beta_3}{3u^3}\right),\\ \phi_{\rm MR}(u)&=\frac1\eta\left(\alpha_0+\alpha_1u -\frac{\alpha_2}{u}+\frac43\alpha_3u^{3/4} +\alpha_4\arctan\frac{u-\alpha_5f_{\rm RD}}{f_{\rm damp}}\right). \end{aligned}\]

The joins are at \(u_1=0.018\) and \(u_2=f_{\rm RD}/2\), in the model regime where \(u_2>u_1\). The reference solves \(\beta_0,\beta_1\) by matching the phase and its \(u\) derivative between inspiral and intermediate at \(u_1\), then solves \(\alpha_0,\alpha_1\) the same way at \(u_2\) (Eq. (37)). This fixes a \(C^1\) piecewise phase. The printed Table 5 \(\beta_1\) and \(\alpha_1\) rows are retained as source evidence; the joined branches use the continuity-determined values instead.

Every fitted row \(\Lambda\in\{\sigma_{1:4},\beta_{1:3},\alpha_{1:5}\}\) uses Eq. (31), with \(x=\chi_{\rm PN}-1\):

\[\Lambda=c_0+c_1\eta+x(c_2+c_3\eta+c_4\eta^2) +x^2(c_5+c_6\eta+c_7\eta^2)+x^3(c_8+c_9\eta+c_{10}\eta^2).\]

The default rows in the reference contain only the digits printed in Table 5. Torch and JAX use inherited extra digits in diffgw_core.fits. Backend checks pass those production rows explicitly into the reference formula. This tests the fit algebra and phase assembly conditional on those digits; it does not establish that the extra digits were published or independently calibrated.

For zero \(t_c\) and \(\phi_c\), the frontends expose two phase conventions. Their coalescence convention returns \(\Psi_c(u)=-\pi/4+\phi(u)-s_{\rm peak}u\). Their reference-frequency convention returns \(\Psi_r(u)=\phi(u)-\phi(u_r)-s_{\rm peak}(u-u_r)\), where \(u_r=Mf_{\rm ref}\). The common slope \(s_{\rm peak}\) is the merger-ringdown derivative of Eq. (13) at the amplitude peak using the raw fitted \(\alpha_1\), not the continuity-determined \(\alpha_1\). This global time gauge is a DiffGW frontend convention, recorded separately from the paper's intrinsic piecewise phase. Both conventions add \(2\pi f t_c-2\phi_c\) when requested.

Component traceability

Component IDs	Paper evidence	Executable reference	Maintained frontend check
`phenomd.phase.fit.{sigma1–4,beta1–3,alpha1–5}.v1`	Eq. (31), Table 5	`fit_expr`, printed rows	Row transcription and Torch/JAX `eval_table_v` algebra for used rows; printed \(\beta_1\) has no native binding because continuity determines the value used by the waveform
`phenomd.phase.pn.v1`	Eqs. (27), (46)–(53)	`pn_phase_coefficients`, `taylorf2_expr`	Torch/JAX TaylorF2 coefficients
`phenomd.phase.inspiral.v1`	Eq. (28)	`inspiral_expr`	Unwrapped phase in inspiral
`phenomd.phase.intermediate.v1`	Eqs. (15)–(16)	`intermediate_shape`, `phase_pieces`	Integrated derivative and unwrapped phase
`phenomd.phase.merger_ringdown.v1`	Eqs. (13)–(14)	`merger_ringdown_shape`, `phase_pieces`	Integrated derivative and unwrapped phase
`phenomd.phase.joins.v1`	Eq. (37)	`phase_pieces`	Symbolic \(C^1\) joins and frontend values around both boundaries
`phenomd.phase.frontend_gauge.v1`	DiffGW convention	`peak_gauge_slope`	Coalescence and reference-frequency phase and frequency gradients

Implementation mapping

COMPONENT_BINDINGS maps each ID to its source, reference function, Torch/JAX entry points and maintained test names. The structural test resolves those links.

Evidence and limitations

The numerical tests cover two selected equal/unequal-mass, aligned-spin cases in FP64, using a declared supplied QNM pair rather than assuming QNM fit provenance. They compare the unwrapped phase on both sides of each join, the reference-frequency anchor, and selected frequency gradients in each branch. These results verify implementation agreement on the stated inputs. They do not establish inherited fit provenance, QNM endpoint parity, complete waveform or polarization agreement, or physical accuracy over the model domain.

Reproduce the checks

Run python -m pytest -q tests/test_imrphenomd_phase_math_reference.py for this slice. The maintained strict mathematical-reference CI gate also collects it with both frontends required.