Higher-mode SPA convention reference

Scope and status

imrphenomhm-spa-conventions/v1, component hm.conventions.quadratic-chirp-spa.v1, is an independent, reference-only diagnostic of a quadratic chirp and its phase conventions. It provides no production change, waveform catalog entry, NR calibration, or physical qualification. Numerical execution results belong in a separate receipt.

Inputs, conventions, and options

Let \(m\) be a positive integer, \(A>0\), \(k>0\), \(\epsilon>0\), and \(\phi(t)=\phi_0+\omega t+kt^2/2\), with real phase parameters. Define

\[h_{\ell m}(t)=C_{\ell m}A e^{-im\phi(t)}e^{-\epsilon t^2/2}, \qquad P_{\ell m}(f)=\int_{-\infty}^{\infty}\overline{h_{\ell m}(t)} e^{-2\pi i f t}\,dt,\quad f>0.\]

\(C_{\ell m}\) is an independently supplied complex constant. Zero coefficients give a zero transform but have no unit phase. The optional paper coefficient phases below assume a positive unequal-mass difference and positive real leading amplitude factors; odd-mode phases are undefined when their amplitude vanishes at equal mass. A required common_td_sign declares the common TD strain sign. No phase or sign is inferred from a waveform comparison.

Mathematics and components

Exact Gaussian and SPA limit

Conjugating \(h_{\ell m}\) gives \(+im\phi_0\). Put \(a=\epsilon-imk\) and \(d=m\omega-2\pi f\). Completing the square gives

\[-\frac a2t^2+idt+im\phi_0 =-\frac a2\left(t-\frac{id}{a}\right)^2-\frac{d^2}{2a}+im\phi_0.\]

The convergent Gaussian integral therefore yields

\[P_{\ell m}(f)=\overline{C_{\ell m}} A e^{im\phi_0}\sqrt{\frac{2\pi}{\epsilon-imk}} \exp\left[-\frac{(m\omega-2\pi f)^2}{2(\epsilon-imk)}\right].\]

The principal square root approaches \(e^{i\pi/4}\sqrt{2\pi/(mk)}\) as \(\epsilon\to0^+\). The stationary point is \(t_*=(2\pi f/m-\omega)/k\), so

\[P^{\rm SPA}_{\ell m}=\overline{C_{\ell m}} A \sqrt{\frac{2\pi}{mk}} e^{i[m\phi(t_*)-2\pi f t_*+\pi/4]}.\]

This limit is exact for the stated oscillatory quadratic-chirp integral. It does not assert exact SPA for a physical inspiral with varying amplitude and chirp rate.

Primitive relative phase and its limitations

For \(u=P/|P|\), \(g=\gcd(2,m)\), \(a_m=2/g\), and \(b_m=m/g\), define

\[T_{\ell m}=u_{\ell m}(f)^{a_m} \overline{u_{22}(2f/m)}^{b_m}.\]

At the matched frequency both chirps have the same stationary point. Their orbital dynamics cancel because \(a_m m=2b_m\):

\[T^{\rm SPA}_{\ell m} =\left(\frac{\overline{C_{\ell m}}}{|C_{\ell m}|}\right)^{a_m} \left(\frac{C_{22}}{|C_{22}|}\right)^{b_m} e^{i(a_m-b_m)\pi/4}.\]

A common orbital shift \(u_{\ell m}\mapsto e^{im\alpha}u_{\ell m}\) and time shift \(u_{\ell m}(f)\mapsto e^{-2\pi if\tau}u_{\ell m}(f)\) cancel. A common FD strain rotation \(e^{i\gamma}\) leaves \(e^{i(2-m)\gamma/g}\); a TD rotation has the conjugate sign. Thus this is not invariant under an arbitrary common strain rotation.

For \((4,4)\), the primitive powers are \((1,2)\) and a mode sign flip is observable. The doubled powers \((2,4)\) erase it. For odd \(m\), the primitive mode power is two, so an odd-mode sign is invisible; a factor \(i\) is visible. Simultaneously flipping all odd-\(m\) modes equals an orbital shift of \(\pi\). Neither this diagnostic nor a \((2,2)\) phase alignment resolves that ambiguity.

Supplied leading-PN coefficient phases

Blanchet et al., arXiv:0802.1249v3, Eqs. (9.3) and (9.4a,b,d,e,h,i), supply these TD unit phases:

| Mode | \(C_{\ell m}/|C_{\ell m}|\) before the common TD sign | | --- | --- | | \((2,2)\) | \(1\) | | \((2,1)\) | \(i\) | | \((3,3)\) | \(-i\) | | \((3,2)\) | \(1\) | | \((4,4)\) | \(-1\) | | \((4,3)\) | \(-i\) |

The transform is derived directly above. Conjugating the entire Fourier-plus transform of the TD mode gives the carrier defined here, including the conjugated coefficient and \(+\pi/4\) contribution.

Consequently \(T_{33}=T_{44}=e^{3\pi i/4}\) for common TD sign \(+1\), and \(e^{-\pi i/4}\) for sign \(-1\). The declared native London-form offsets \(\Delta_{\ell m}=\pi[3\ell+(\ell+m)\bmod2-2]/2\), entering as \(e^{-i\Delta_{\ell m}}\) (London et al., v4, Eq. 7), instead give \(T_{33}=T_{44}=-1\).

Choosing the principal angle of \(C_{22}\), then removing \(m/2\) times the SPA 22 carrier phase \(-\arg C_{22}+\pi/4\), preserves the \((2,2)\) carrier. The resulting phase correction from the London-form baseline is, modulo \(2\pi\),

\[\delta\Phi_{\ell m}=\begin{cases} (1-m/2)\pi/4,&\text{common TD sign }+1,\\ -(1-m/2)3\pi/4,&\text{common TD sign }-1. \end{cases}\]

Changing the \(C_{22}\) angle branch by \(2\pi\) changes this orbital alignment by \(\pi\), flipping every odd-\(m\) carrier. The primitive invariants remain unchanged. This conditional mathematical difference alone selects neither a native production convention nor an NR conversion.

For sign \(-1\), the multiplier decomposes exactly as \(e^{i(m/2-1)K}=e^{-iK}e^{imK/2}\) with \(K=3\pi/4\): a common positive-frequency carrier phase and an orbital rotation. This identity does not by itself prove a native defect or a physical polarization-gauge equivalence. Real polarizations require conjugate factors at negative frequency; multiplication of all positive frequency carriers by a phase is not automatically a real polarization rotation. Signed-mode projection and the intended gauge must be checked before production use. The signed-pair extension derives the conjugate transport of both Fourier contributions and distinguishes it from a polarization-basis rotation. Its execution status is recorded separately.

Implementation mapping

src/diffgw_reference/imrphenomhm_spa_conventions.py contains the symbolic Gaussian/SPA expressions, independent scalar evaluations, primitive invariant, and optional paper phases. There are no Torch/JAX bindings or production imports. tests/test_imrphenomhm_spa_conventions_math_reference.py maintains symbolic proofs, direct TD quadrature, damping-limit sequences, gauge tests, and sign-observability checks.

Evidence and limitations

The quadrature checks a finite-damping TD integrand against the exact full-line expression, with a separately bounded omitted Gaussian tail. The five damping values \(10^{-2},10^{-4},10^{-6},10^{-8},10^{-10}\) check the complex SPA limit, including magnitude, coefficient phase, initial phase, and square-root branch. All six paper coefficient phases are checked against the orbital-normalized London-form difference. These checks have no NR input and do not compare either native waveform backend with a physical inspiral. The separate SXS TD/FD convention receipt records the basis conversion, raw-data sign check and fixed-conversion diagnostic.

Reproduce the checks

On the designated compute host, with the project and optional reference dependencies installed:

python -m pytest -q tests/test_imrphenomhm_spa_conventions_math_reference.py

The filename joins the maintained strict math-reference inventory automatically. --require-math-reference requires the complete inventory, so it is not a focused-module flag.