Higher Multipolar Modes

The dominant $(2,\pm2)$ quadrupole does not exhaust a compact-binary waveform. Modes with other $(\ell,m)$ encode mass asymmetry, spin-dependent structure, and angular features that can become prominent near merger or away from a face-on line of sight. Whether a mode is essential is therefore conditional: it depends on source symmetry, intrinsic scale, observer orientation, and the physical question being asked.

Helpful background. Einstein Equations supplies the nonlinear source of multipoles; Waves, Green Functions, and Geometric Optics supplies outgoing-wave phases; Relativistic Two-Body Dynamics supplies the accepted orbital history; Gravitational Radiation and Waveforms supplies radiative moments and harmonics; and Computational Relativity supplies strong-field mode data near merger.

Modal content and normalization

At source-frame areal radius $r$, use

$$H=r(h_+-ih_\times) =\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} H_{\ell m}\,{}_{-2}Y_{\ell m},$$

with unit-normalized spin-weight $-2$ harmonics and time dependence $e^{-i\omega t}$. In a nonprecessing quasi-circular source frame,

$$H_{\ell m}(t) =A_{\ell m}(x,\nu,\delta,\boldsymbol\chi_A) e^{-im\psi(t)}, \qquad \omega_{\ell m}\simeq m\Omega.$$

Here $x=(GM\Omega)^{2/3}$, $\nu=m_1m_2/M^2$, and $\delta=(m_1-m_2)/M$. A source-frame distance-rescaled mode $H_{\ell m}^{\rm src}$ has dimensions of length. At cosmological redshift its detector-facing contribution is

$$\frac{H_{\ell m}^{\rm det}}{D_L} =\frac{(1+z)H_{\ell m}^{\rm src}}{D_L},$$

equivalently with every dimensional source mass replaced by its redshifted value. The nearby-source limit is $H_{\ell m}/D$.

Unit harmonic normalization makes the total luminosity additive before mode interference is projected onto a particular direction:

$$\mathcal F =\frac{1}{16\pi G} \sum_{\ell,m}\left\lvert\dot H_{\ell m}\right\rvert^2.$$

By contrast, an observer sees the coherent sum of modes at one $(\theta,\phi)$, so amplitudes and phases can reinforce or cancel. A small fraction of angle-integrated power can still make an appreciable directional correction.

Symmetry before amplitude counting

For a planar nonprecessing binary in the standard source frame, reflection symmetry gives

$$H_{\ell,-m}=(-1)^\ell\overline{H}_{\ell m}.$$

This halves the independent mode data and supplies a sharp convention check. For an equal-mass, nonspinning circular binary, a half-turn combined with exchange of the bodies also forces odd-$m$ modes to vanish. Unequal masses break that exchange symmetry, and their leading odd-$m$ amplitudes are proportional to $\delta$. Equal masses with unequal spins need not retain the same suppression.

The circular limit does not say that only $\ell=2$ exists. Even-$m$ modes such as $(3,2)$ and $(4,4)$ can be nonzero, while their leading slow-motion powers make them smaller early in inspiral. Schematically,

$$\frac{H_{33}}{H_{22}}\sim\delta x^{1/2}, \qquad \frac{H_{44}}{H_{22}}\sim x,$$

up to convention-dependent numerical coefficients and spin corrections. These ratios grow with frequency, which explains why subdominant modes are generally more relevant late in the signal.

The mode and parity relations, including their dependence on the chosen frame, are developed in Blanchet 2024, §3.4.5, pp. 212–219 and Boyle et al. 2014, §§II and IV–V.

Orientation and physical materiality

At the north pole of the source frame,

$${}_{-2}Y_{\ell m}(0,\phi) =\sqrt{\frac{2\ell+1}{4\pi}}\, \delta_{m2}\,e^{2i\phi}.$$

Thus a face-on observer samples only $m=2$ from the displayed $H$ convention, even though the source radiates other $m$ values. Away from the axis—especially near an edge-on orientation—additional harmonics are nonzero and the waveform develops richer amplitude and phase structure.

In particular, face-on does not mean quadrupole only: $(3,2)$, $(4,2)$, and other $m=2$ modes can survive. Conversely, the dominant quadrupole does not vanish edge-on. At leading order its plus amplitude is proportional to $1+\cos^2\iota$, whereas its cross amplitude is proportional to $2\cos\iota$; at $\iota=\pi/2$ the cross contribution vanishes but the plus contribution remains. Higher-mode importance follows from comparing the complete projected strain, not from assuming the $(2,2)$ signal disappears.

Four conditions make higher modes more likely to be physically material:

1. Mass asymmetry: large $q=m_1/m_2$ makes $\delta$ large and lifts odd-$m$ suppression.
2. Inclination: an off-axis line of sight exposes harmonics hidden at the pole.
3. Late frequency: increasing $x$ weakens the early-inspiral PN hierarchy.
4. Spin structure: aligned spins alter intrinsic amplitudes, while precession rotates and mixes modes in an inertial frame.

“Material” must still name a criterion. For radiated energy, compare $\sum|\dot H_{\ell m}|^2$. For waveform morphology, compare the coherent strain along a fixed line of sight. For a quantitative data-analysis standard, the noise weighting and inference question belong to gravitational-wave astronomy rather than this source-independent construction.

Frames, mixing, and interpretation

A rotation changes mode coefficients through a Wigner matrix:

$$H^{\mathrm{inertial}}_{\ell m} =\sum_{m'=-\ell}^{\ell} D^\ell_{mm'}(R)\, H^{\mathrm{source}}_{\ell m'}.$$

It mixes $m$ values at fixed $\ell$ but cannot create a new physical degree of freedom. A precessing waveform can therefore have complicated inertial-frame sidebands even when its coprecessing-frame content remains dominated by $(2,\pm2)$. Comparing mode amplitudes without aligning frames can misidentify a rotation as enhanced intrinsic radiation.

Spherical modes can also mix with spheroidal remnant modes near ringdown, but that is a basis mismatch rather than source mass asymmetry. The defining basis, frame, retarded time, and polarization dyad must accompany every quoted $H_{\ell m}$.

Detailed binary equations of motion, production numerical algorithms, and observational parameter estimation are not treated here. They respectively supply the source evolution, strong-field waveform data, and empirical criterion to which this modal structure may be applied.

Common pitfalls

Equating “subdominant” with negligible. The label usually refers to an intrinsic amplitude or angle-integrated power hierarchy. Directional suppression of $(2,2)$ can change the observed hierarchy.

Applying equal-mass symmetry when spins break exchange. Odd-$m$ cancellation requires the full binary configuration, not masses alone, to be invariant under the relevant exchange and rotation.

Comparing inertial and coprecessing coefficients directly. A Wigner rotation redistributes the same radiation among $m$ values. Frame alignment must precede any claim of intrinsic mode enhancement.

Exercise

An equal-mass, nonspinning circular binary has nonzero coprecessing modes $H_{22}$, $H_{32}$, and $H_{44}$. Which of these can contribute for an exactly face-on observer? What changes for a slightly unequal-mass edge-on binary?

Solution

At $\theta=0$, the spin-weighted harmonic identity selects $m=2$. Therefore $H_{22}$ and $H_{32}$ can contribute, whereas $H_{44}$ vanishes along that exact direction despite being present in the source radiation. For equal masses, every odd-$m$ mode is additionally forbidden by exchange symmetry, but that does not eliminate $H_{32}$ because $m=2$. Making the masses unequal activates odd-$m$ modes with leading amplitudes proportional to $\delta$. Moving to an edge-on line of sight makes many more spin-weighted harmonics nonzero, so both the newly activated odd-$m$ content and modes such as $H_{44}$ can alter the coherent strain.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024) 4. DOI. Open PDF.
• Boyle, Michael, et al. “Gravitational-Wave Modes from Precessing Black-Hole Binaries.” Physical Review D 90 (2014) 124010. DOI. Open PDF.