Ringdown and quadratic QNM models (tgr.nrsurqnm)

The tgr.nrsurqnm module builds beyond-GR ringdown waveforms on top of the numerical-relativity surrogate NRSur7dq4. The full inspiral-merger-ringdown signal is generated with the surrogate, and the late-time ringdown of selected spherical-harmonic modes is then re-modelled as a superposition of quasi-normal modes (QNMs). Working at the level of the QNMs allows controlled, physically meaningful deviations from General Relativity (GR) to be injected mode by mode, which is exactly what a test of GR requires.

This page describes the physics and the mathematics implemented by the module. The auto-generated API reference for every function lives under tgr.nrsurqnm in the tgr package package documentation.

Quasi-normal-mode ansatz

After merger, each spherical-harmonic mode of the strain rings down as a sum of damped sinusoids. A single QNM labelled by \((\ell, m, n)\) (harmonic indices and overtone number \(n\)) is

(1)\[\psi_{\ell m n}(t) = A_{\ell m n}\, \exp\!\big[-i\,\omega_{\ell m n}\,(t - t_0)\big], \qquad t \ge t_0 ,\]

where \(t_0\) is the ringdown start time and \(A_{\ell m n}\in\mathbb{C}\) is a complex amplitude that fixes both the size and the initial phase of the mode. The complex angular frequency packages the oscillation frequency \(f_{\ell m n}\) and the damping time \(\tau_{\ell m n}\) into a single quantity,

(2)\[\omega_{\ell m n} = 2\pi f_{\ell m n} - \frac{i}{\tau_{\ell m n}} ,\]

so that the real part drives the oscillation and the imaginary part produces the exponential decay,

\[\psi_{\ell m n}(t) = A_{\ell m n}\, e^{-i\,2\pi f_{\ell m n}(t-t_0)}\; e^{-(t-t_0)/\tau_{\ell m n}} .\]

QNM spectrum of the remnant

The QNM frequencies and damping times are uniquely determined by the mass \(M_f\) and dimensionless spin \(\chi_f\) of the remnant black hole. The helper tgr.nrsurqnm.get_qnmpar() first predicts \((M_f, \chi_f)\) from the initial binary parameters using the NRSur7dq4 remnant fits, and then maps them to \((f_{\ell m n}, \tau_{\ell m n})\) for each requested mode.

Two kinds of mode labels are supported:

• Linear (fundamental/overtone) modes — a 3-character label "lmn" such as "220" or "221", evaluated directly as \((f_{\ell m n}, \tau_{\ell m n})\).

• Quadratic modes (QQNMs) — a 6-character label that concatenates two linear modes, e.g. "220220" for \(220\times220\). A quadratic mode arises from the second-order coupling of two parent linear modes and rings at the sum frequency with the combined decay rate,

  (3)\[f_{(1)(2)} = f_{(1)} + f_{(2)} , \qquad \frac{1}{\tau_{(1)(2)}} = \frac{1}{\tau_{(1)}} + \frac{1}{\tau_{(2)}} .\]

Least-squares QNM decomposition

Given a target waveform \(h(t)\) and a chosen set of \(M\) modes, tgr.nrsurqnm.qnm_decomposition() extracts the complex amplitudes by a linear least-squares fit over the ringdown window \(t \in [t_0,\, t_0 + T]\), where the window length \(T = \log(1000)\,\max_k \tau_k\) is set by the slowest-decaying mode (i.e. it extends until the dominant amplitude has dropped by a factor of \(10^3\)).

Sampling the templates (1) (with unit amplitude and an overall dynamical-range factor \(\eta=10^{-22}\) for numerical conditioning) on the \(N\) time samples \(t_j\) defines the design matrix \(G\in\mathbb{C}^{N\times M}\),

\[G_{jk} = \eta\,\exp\!\big[-i\,\omega_{k}\,(t_j - t_0)\big] .\]

The amplitudes \(A = (A_1,\dots,A_M)^\top\) are the minimiser of the residual \(\lVert G A - h \rVert_2^2\), i.e. the solution of the normal equations

(4)\[\big(G^{H} G\big)\, A = G^{H} h , \qquad A = \big(G^{H} G\big)^{-1} G^{H} h ,\]

where \(G^{H}\) denotes the conjugate transpose. The fitted amplitudes are finally rescaled by \(\eta\) and propagated to the requested reference time \(t_0\) through the phase factor \(e^{-i\omega_k (t_0 - t_{\rm fit})}\).

Replacing the (4,4) ringdown: gen_nrsurqnm

tgr.nrsurqnm.gen_nrsurqnm() generates the full NRSur7dq4 signal, keeps every mode except \((4,\pm 4)\), and reconstructs the \((4,4)\) ringdown from its QNM decomposition (4). Deviations from GR are injected through per-mode fractional amplitude parameters \(\delta_{\ell m n}\) (keyword arguments delta_<mode>):

(5)\[A_{\ell m n} \;\longrightarrow\; A_{\ell m n}\,\big(1 + \delta_{\ell m n}\big), \qquad \delta_{\ell m n}=0 \ \text{recovers GR.}\]

The reconstructed mode is mapped back to the two polarizations with the spin-weighted spherical harmonics of weight \(-2\) (see Recombination into polarizations).

Removing quadratic modes: gen_nrsur_remove_qqnm

tgr.nrsurqnm.gen_nrsur_remove_qqnm() isolates the contribution of the quadratic modes to the \((4,4)\) ringdown so that a GR deviation can be applied to them. The amplitude of a quadratic mode is fixed by the amplitudes of its two parent linear \((2,2)\) modes, a theory-predicted complex ratio \(R_{(1)(2)}(\chi_f)\) (interpolated as a function of remnant spin), and a unit conversion to the surrogate’s dimensionless strain,

(6)\[A_{(1)(2)} = A_{(1)}\,A_{(2)}\,R_{(1)(2)}(\chi_f)\,\mathcal{C}, \qquad \mathcal{C} = \frac{d_L\,(\mathrm{Mpc}\!\to\!\mathrm{m})} {M_f\,\big(M_\odot\!\to\!\mathrm{m}\big)} ,\]

where \(d_L\) is the luminosity distance and \(\mathcal{C}\) rescales the physical amplitudes back to surrogate units. Each quadratic mode is then built as a damped sinusoid (1) with parameters (3) and subtracted from the \((4,4)\) ringdown.

Amplitude and phase deviation. The subtracted quadratic contribution is scaled by a single complex deviation factor

(7)\[\kappa = a_{\rm TGR}\,e^{\,i\,\varphi_{\rm TGR}} ,\]

controlled by the keyword arguments quadratic_tgr (amplitude \(a_{\rm TGR}\)) and quadratic_tgr_phase (phase \(\varphi_{\rm TGR}\), in radians). The GR limit is \(a_{\rm TGR}=1,\ \varphi_{\rm TGR}=0\), i.e. \(\kappa = 1\), which removes exactly the GR-predicted quadratic mode. Tuning \(a_{\rm TGR}\) rescales how much of the mode is removed, while \(\varphi_{\rm TGR}\) rotates it in the complex plane, probing a phase offset of the quadratic coupling.

Frequency and damping-time deviation. Optionally, a quadratic mode with non-GR spectral parameters can be re-injected. The keyword arguments qqnm_deltaf and qqnm_deltatau shift the spectrum fractionally,

(8)\[f \;\longrightarrow\; f\,(1 + \delta f), \qquad \tau \;\longrightarrow\; \tau\,(1 + \delta\tau),\]

so the residual \((4,4)\) ringdown retains a quadratic mode that oscillates and decays at shifted rates relative to the GR prediction.

Recombination into polarizations

A mode \((\ell, m)\) in the co-precessing/source frame is projected onto the observer frame using the spin-weight \(-2\) spherical harmonics \({}_{-2}Y_{\ell m}(\iota, \phi)\), evaluated at the inclination \(\iota\) and a reference phase \(\phi = \pi/2 - \phi_{\rm coa}\). The modelled \((4,4)\)/\((4,-4)\) ringdown is recombined as

(9)\[h_{44}^{\rm obs}(t) = h_{44}(t)\, {}_{-2}Y_{4,4}(\iota,\phi) + h_{44}^{*}(t)\, {}_{-2}Y_{4,-4}(\iota,\phi) ,\]

and the complex strain \(h = h_+ - i\,h_\times\) finally yields the two polarizations returned by every generator, \(h_+ = \mathrm{Re}\,h\) and \(h_\times = -\,\mathrm{Im}\,h\).

Tapered surrogate helper

For convenience tgr.nrsurqnm.gen_nrsur7dq4_tdtaper() returns the plain NRSur7dq4 polarizations with a short Tukey-style taper of length window (default \(0.05\,\mathrm{s}\)) applied at the start of the time series to suppress turn-on transients.