Gravitational Wave Modes and Spherical Harmonics

Overview

Spherical Harmonics ($Y_{\ell m}(\theta, \phi)$)

Spherical harmonics are solutions to the angular part of Laplace's equation in spherical coordinates. They are expressed as functions of two angles, $\theta$ (colatitude) and $\phi$ (longitude), and are represented as:

$$ Y_{\ell m}(\theta, \phi) = \sqrt{\frac{(2\ell+1)}{4\pi} \frac{(\ell-m)!}{(\ell+m)!}} P_{\ell}^m(\cos\theta) e^{im\phi}, $$

where:

• $ell$ is the degree, determining the overall scale of variation.
• $m$ is the order, representing azimuthal variations.
• $P_{\ell}^m(\cos\theta)$ are the associated Legendre polynomials.

These functions form an orthonormal basis for functions defined on the surface of a sphere and are widely used in quantum mechanics, geophysics, and astrophysics to describe angular distributions.

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Spin-Weighted Spherical Harmonics ($Y^{s}_{\ell m}(\theta, \phi)$)

Spin-weighted spherical harmonics generalize the concept of spherical harmonics to describe fields with intrinsic spin, such as the polarization of light or the gravitational wave strain. These functions incorporate a spin-weight $s$, which represents how the field transforms under rotations.

The spin-weighted spherical harmonics are derived by applying spin-raising or spin-lowering operators to the standard spherical harmonics:

$$ Y_{s\ell m}(\theta, \phi) = \sqrt{\frac{(2\ell+1)}{4\pi}} D^\ell_{m, -s}(\phi, \theta, 0), $$

where $D^\ell_{m, -s}$ are the Wigner $D$-functions, describing rotations in quantum mechanics.

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Connection to Gravitational Waves

Gravitational waves are ripples in spacetime caused by massive, accelerating objects, such as merging black holes or neutron stars. These waves are typically decomposed into modes defined on a spherical coordinate system centered on the source. Spin-weighted spherical harmonics ($s = -2$) are used to describe these waves because the gravitational wave strain is a spin-2 tensor field.

Decomposition of Gravitational Waves

Gravitational wave signals are often expanded into spin-weighted spherical harmonic modes:

$$ h(t, \theta, \phi) = \sum_{\ell=2}^\infty \sum_{m=-\ell}^\ell h_{\ell m}(t) , {-2}Y{\ell m}(\theta, \phi), $$

where $h_{\ell m}(t)$ are the mode amplitudes, encoding the time-dependent information of the wave. This decomposition separates angular and temporal components, making it easier to analyze and visualize.

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Higher-Order Modes (HOM) and Their Importance

HOM of gravitational waves contribute beyond the dominant quadrupole ((\ell=2, m=\pm2)) mode, especially in asymmetric systems. They provide:

• Enhanced sensitivity to source properties, such as mass ratio, spin orientations, and orbital eccentricities.
• Deeper insights into binary black hole merger dynamics.
• Improved parameter estimation and detectability of distant sources.

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Conclusion

Spherical harmonics and spin-weighted spherical harmonics are essential for visualizing and analyzing gravitational waves. Their ability to describe angular structures and encode spin transformations enriches our understanding of astrophysical phenomena and aids in interpreting gravitational wave data.

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Published work:

• Kashi, B., Visualization of Gravitational Radiation from Binary Black Holes (2024 ASI Conference Proceedings)
• Visualization of Gravitational Radiation from Binary Black Holes (ResearchGate)
• Visualising Higher-Order Gravitational Radiation Modes in Binary Black Hole Spacetimes (ResearchGate)

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The above information is documented in vis_readme.pdf