IMRPhenomD amplitude and joins, first reference slice

Scope and status

imrphenomd-amplitude-joins-paper-precision/v1 is an executable specification in src/diffgw_reference/imrphenomd_amplitude.py. Its primary source is Khan et al., Paper II, arXiv:1508.07253v2. This slice specifies the dominant-mode scaled amplitude and seven printed calibration rows. It takes the total-mass-unit ringdown frequency \(f_{\rm RD}\) and damping frequency \(f_{\rm damp}\) as inputs. Remnant fits, Kerr QNMs, phase, polarizations, the \(Mf=0.2\) implementation cutoff and optional tapers are outside this version's mathematical claim. The remnant reference slice specifies the remnant fits and conditional QNM inputs used at this boundary. The phase reference slice specifies the intrinsic phase branches and their joins.

Inputs, conventions, and options

The dimensionless frequency is \(u=Mf\), with \(M\) in seconds when \(f\) is in Hz. The amplitude below is \(A/A_0\); the common Newtonian factor \(A_0\propto f^{-7/6}\) is restored by the frontends. For masses \(m_1\geq m_2>0\), \(\eta=m_1m_2/M^2\), \(\delta=(m_1-m_2)/M\), \(\chi_s=(\chi_1+\chi_2)/2\), and \(\chi_a=(\chi_1-\chi_2)/2\).

Mathematics and components

Paper II Eqs. (2)–(4) give

\[\chi_{\rm PN}=\frac{m_1\chi_1+m_2\chi_2}{M} -\frac{38\eta}{113}(\chi_1+\chi_2).\]

Table 5 uses this unscaled \(\chi_{\rm PN}\), not the normalized \(\hat\chi\) discussed elsewhere in the paper. Its seven amplitude rows \(\rho_{1,2,3},v_2,\gamma_{1,2,3}\) are transcribed as exact decimal strings, using only the digits printed in Table 5. Each row \(c_0,\ldots,c_{10}\) enters Eq. (31), with \(x=\chi_{\rm PN}-1\):

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

Eqs. (29)–(30) and Appendix B Eqs. (54)–(60) give the inspiral branch

\[a_{\rm ins}(u)=\sum_{k=0}^6\mathcal A_k(\pi u)^{k/3} +\rho_1u^{7/3}+\rho_2u^{8/3}+\rho_3u^3.\]

The seven \(\mathcal A_k\) are written as exact rational and symbolic \(\pi\) terms in the reference module. Eq. (19) gives

\[a_{\rm MR}(u)=\frac{\gamma_1\gamma_3f_{\rm damp}} {(u-f_{\rm RD})^2+(\gamma_3f_{\rm damp})^2} \exp\left[-\frac{\gamma_2(u-f_{\rm RD})}{\gamma_3f_{\rm damp}}\right].\]

The lower join is \(u_1=0.014\); the upper join \(u_3\) is the peak in Eq. (20) where \(|\gamma_2|<1\). The existing Torch/JAX implementations continue that formula at \(|\gamma_2|\geq1\) by setting the square root to zero. That gives a real join frequency but not a stationary point of Eq. (19); the reference names this separately as peak_compatibility and uses the true derivative \(a'_{\rm MR}(u_3)\) in the join. This continuation is an implementation choice, not a result established by Eq. (20).

For \(t=(u-u_1)/(u_3-u_1)\) and \(h=u_3-u_1\), the middle branch is the unique quartic matching \(a_{\rm ins}\) and its derivative at \(u_1\), \(a_{\rm MR}\) and its derivative at \(u_3\), and fitted \(v_2\) at \(u_2=(u_1+u_3)/2\) (Eqs. (21)–(26), Table 2). The reference constructs it as a cubic Hermite interpolant \(H(t)\) plus \(16[v_2-H(1/2)]t^2(1-t)^2\). This differs algebraically from the frontend Vandermonde solve but defines the same quartic. Both branches and their first derivatives therefore meet at each join.

Implementation mapping

The backend bindings are torchwave/approximants/imrphenomd.py and jaxwave/approximants/imrphenomd.py: eval_table_v evaluates the fit IDs, pn_amplitude_coefficients the PN ID, amplitude_peak_frequency the peak ID, and intermediate_amplitude_coeffs with compute_slope_peak the join ID. forward (Torch) and evaluate (JAX) assemble the spin coordinates and three amplitude branches. The maintained test_imrphenomd_math_reference.py checks every mapped component, including values on both sides of each join and frequency derivatives in all three open regions.

Evidence and limitations

Evidence and checks

Component IDs	Paper anchor	Comparison
phenomd.amp.fit.*.v1	Eq. (31), Table 5	Printed-row transcription and hierarchical evaluation on both frontends
phenomd.amp.spin_coordinates.v1	Eqs. (2)–(4)	Exact spin coordinate construction in test cases
phenomd.amp.pn.v1, phenomd.amp.inspiral.v1	Eqs. (29)–(30), (54)–(60)	All PN coefficients and inspiral waveform-amplitude ratios
phenomd.amp.merger_ringdown.v1, phenomd.amp.peak.v1	Eqs. (19)–(20)	Peak formula and merger-ringdown amplitude ratios; continuation called out separately
phenomd.amp.intermediate.v1	Eqs. (21)–(26), Table 2	Exact five-condition identity, interior values and endpoint slopes against Torch/JAX

The production TABLE_V_UNROUNDED_COEFFS stores additional inherited digits whose publication provenance remains unresolved. Tests pass that table explicitly to the independent Eq. (31) formula to verify implementation algebra, while the reference's own default rows contain only published digits. Agreement under the supplied production rows is not evidence that the extra digits appear in the paper. The complete amplitude test compares ratios across all three regions, canceling the shared SI conversion and orientation factors. It qualifies the amplitude equations conditional on the supplied remnant frequencies and calibration data; it does not qualify the complete waveform or its physical accuracy.

Reproduce the checks

Run pytest -q tests/test_imrphenomd_math_reference.py. The maintained strict --require-math-reference gate also collects this module automatically.