⚛️ general relativity

Determination of the Angular Momentum of Radiated Gravitons, Scalars, and Dark Photons from Binary Orbits

Using a field-theoretic approach, this paper demonstrates that compact binaries radiate angular momentum such that each emitted graviton carries approximately $2\hbar$ while scalar and dark photon quanta carry approximately $\hbar$ , a result that holds for both elliptical and hyperbolic orbits and can be determined from orbital evolution despite the impossibility of a gauge-invariant decomposition into spin and orbital components.

Original authors: Arpan Hait, Subhendra Mohanty

Published 2026-08-05

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Arpan Hait, Subhendra Mohanty

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic dance floor where massive objects like stars and black holes spin and orbit each other. Sometimes, these dancers move in perfect circles, but often they trace out stretched-out ovals (ellipses) or even swing past each other on wild, open curves (hyperbolas). As they dance, they don't just move through space; they shake the very fabric of the universe itself. This shaking creates ripples called gravitational waves, which are like sound waves but made of spacetime. Scientists have known for a long time that these ripples carry away energy, causing the dancers to spiral closer together. But there's a deeper mystery: exactly what is being carried away? Is it just energy, or is it also a specific type of "twist" called angular momentum? And if these ripples are made of tiny particles (like photons are made of light), how much "twist" does each individual particle carry? This question is crucial because the amount of twist a particle carries is directly linked to its "spin," a fundamental property that tells us what kind of particle it is. If we can measure this twist, we might be able to prove that gravity is carried by a specific type of particle called a graviton, and even detect invisible particles like "dark matter" that might be hiding in these cosmic dances.

This paper takes a deep dive into that mystery, acting like a cosmic accountant for the universe's most energetic dance moves. The authors, Arpan Hait and Subhendra Mohanty, use a sophisticated mathematical toolkit called "field theory" to count the particles being emitted by binary systems (two objects orbiting each other) and measure the angular momentum each one carries. They looked at three types of radiation: the familiar gravitational waves, and two hypothetical types of radiation from invisible particles called "scalars" and "dark photons" (vector particles).

Here is the big discovery: The authors found a clear, distinct signature for each type of particle. When a binary system emits gravitational waves, every single graviton particle it shoots out carries away exactly 2ℏ (two times the fundamental unit of quantum twist) of angular momentum. This holds true whether the stars are dancing in a neat oval or swinging past each other on a wild hyperbolic path. It's as if every graviton is a tiny top spinning with a specific, double-strength twist.

However, the story changes if the universe is leaking invisible particles. If the binary system is radiating scalar particles (like a type of fuzzy dark matter) or vector particles (like dark photons), the math shows something different. In these cases, each emitted particle carries away only 1ℏ of angular momentum. It's a single twist instead of a double twist.

The paper also tackles a tricky problem: can we tell the difference between the "spin" of the particle and the "orbital" motion of the system? The authors show that while the total amount of twist is measurable, we cannot separate it into "spin" and "orbit" parts in a way that makes sense to every observer (it's not "gauge-invariant"). We can only measure the total. But that's okay! Because the total twist per particle is so different for gravitons (2ℏ) compared to scalars and vectors (1ℏ), we can use this as a fingerprint.

By looking at how the orbit of a binary system changes over time—specifically how fast it loses energy and how its shape (eccentricity) changes—astronomers can calculate the ratio of angular momentum lost to the number of particles emitted. If that ratio is close to 2ℏ, it confirms the radiation is gravitational. If it drops to 1ℏ, it suggests the system is leaking scalar or vector particles. The authors even checked this for "hyperbolic" encounters, where two black holes swing past each other once and never return, and found the same rule applies: the ratio settles at 2ℏ for gravitons.

In short, this paper suggests that by carefully watching how cosmic dancers lose their spin, we might finally catch a glimpse of the invisible particles that make up the dark side of the universe, distinguishing them from the gravity we already know. It turns the abstract math of quantum fields into a practical way to count the invisible particles shaking the cosmos.

Technical Summary: Determination of the Angular Momentum of Radiated Gravitons, Scalars, and Dark Photons from Binary Orbits

Problem Statement
The paper addresses the characterization of angular momentum radiated by compact binary systems through gravitational waves, as well as potential scalar and vector (dark photon) fields. While the energy loss of binary orbits due to gravitational radiation is well-established (Peters-Mathews formula), the specific angular momentum carried away per emitted quantum has received less attention. Recent work by Page [1] noted that for elliptical orbits, the ratio of radiated angular momentum to the number of emitted gravitons ( $\dot{J}/\dot{N}$ ) is approximately $2\hbar$ . This paper seeks to extend this analysis to hyperbolic orbits and to scalar and vector radiation fields. A central theoretical challenge addressed is the gauge-invariant decomposition of the total angular momentum into spin and orbital components; while the total angular momentum is gauge-invariant, its individual spin and orbital parts are not for spin-1 and spin-2 fields. The authors aim to determine if the ratio $\dot{J}/\dot{N}$ can serve as an observable signature to distinguish the spin of the radiated particles (spin-2 for gravitons vs. spin-0/1 for scalars/vectors) and to constrain ultralight dark matter candidates.

Methodology
The authors employ a field-theoretic approach to compute energy and angular momentum fluxes.

Framework: They treat the binary source as a classical current (stress-energy tensor $T^{\mu\nu}$ for gravitons, baryon number density $n(x)$ for scalars, and vector current $J^\mu$ for dark photons) while quantizing the radiated fields.
Formalism: Using Fermi's Golden Rule and the interaction vertices derived from the Lagrangian, they calculate the emission rates in the frequency domain. The source currents are expanded in Fourier series (for elliptical orbits) or Fourier transforms (for hyperbolic encounters).
Approximations: The quadrupole approximation is used for gravitational radiation ( $e^{-i\mathbf{k}\cdot\mathbf{x}} \approx 1$ ), valid for non-relativistic compact binaries. For scalar and vector radiation, the dipole approximation is utilized, retaining terms linear in $\mathbf{k}\cdot\mathbf{x}$ .
Orbital Dynamics: Calculations are performed for both elliptical orbits (parametrized by eccentricity $e$ and fundamental frequency $\omega_0$ ) and hyperbolic encounters (parametrized by hyperbolic eccentricity $e > 1$ ).
Quantities Computed: The authors derive expressions for the rate of energy radiated ( $\dot{E}$ ), the rate of particle number emission ( $\dot{N}$ ), and the rate of total angular momentum radiated ( $\dot{J}$ ). They specifically compute the ratio $\dot{J}/\dot{N}$ for individual frequency modes and integrated over all modes.

Key Contributions and Results

Gravitational Radiation (Elliptical Orbits):
- The authors reproduce the classical results for energy and angular momentum loss, confirming the independence of these rates from $\hbar$ .
- They calculate the angular momentum per graviton ( $\dot{J}/\dot{N}$ ) mode-by-mode. For the fundamental mode ( $n=2$ ) in circular orbits, the ratio is exactly $2\hbar$ .
- For higher harmonics ( $n > 2$ ) and non-zero eccentricity, the ratio deviates from $2\hbar$ at low $n$ but saturates to $2\hbar$ as $n \to \infty$ .
- The total angular momentum per graviton, integrated over all modes, remains approximately $2\hbar$ across a wide range of eccentricities, though the paper notes it is technically eccentricity-dependent and the ratio approaches $2\hbar$ more closely as eccentricity increases.
- The authors decompose the total angular momentum into orbital ( $\dot{L}$ ) and spin ( $\dot{S}$ ) components, finding $\dot{L}/\dot{N} \simeq (2/3)\hbar$ and $\dot{S}/\dot{N} \simeq (4/3)\hbar$ . They emphasize that while the sum is gauge-invariant, the individual components are not.
Gravitational Radiation (Hyperbolic Orbits):
- Extending the analysis to unbounded hyperbolic encounters, the frequency spectrum is continuous ( $\omega' = \nu\omega_0$ ).
- The ratio $\dot{J}/\dot{N}$ for individual frequency modes starts near zero at low frequencies (associated with the gravitational wave memory signal) and approaches $2\hbar$ at high frequencies.
- The total integrated ratio $\Delta J / \Delta N$ for the entire encounter is a function of the hyperbolic eccentricity; it remains close to $2\hbar$ for smaller eccentricities and approaches $2\hbar$ in the limit of large hyperbolic eccentricity.
Scalar and Vector Radiation:
- The authors compute radiation from ultralight scalar fields (e.g., axions) and vector fields (dark photons) coupled to baryon number or charge.
- Unlike the quadrupolar nature of gravitational sources, scalar and vector sources are dominated by the dipole moment.
- For both scalar and vector radiation, the ratio of total angular momentum to particle number is found to be $\dot{J}/\dot{N} \simeq \hbar$ for $e < 1$ .
- This ratio is not strictly independent of eccentricity; it decreases with increasing eccentricity and approaches zero in the limit $e \to 1$ , as the angular momentum loss vanishes in this limit.

Significance and Claims
The paper claims that the ratio of radiated angular momentum to the number of emitted quanta ( $\dot{J}/\dot{N}$ ) serves as a distinct signature for the spin of the radiated field:

Spin-2 (Gravitons): The ratio is $\approx 2\hbar$ , reflecting the quadrupolar nature of the source.
Spin-0 and Spin-1 (Scalars/Vectors): The ratio is $\approx \hbar$ , reflecting the dipolar nature of the source.

The authors argue that while binary orbital dynamics (specifically the evolution of semi-major axis $a$ and eccentricity $e$ ) can constrain the existence of extra long-range fields and determine the sign of the coupling parameter $\alpha$ (distinguishing between scalar and vector mediated forces), they cannot uniquely prove the quantization of the graviton's spin. This is because observations only yield the total angular momentum loss, which is the sum of spin and orbital components, neither of which is individually gauge-invariant. However, the distinct values of $\dot{J}/\dot{N}$ ( $2\hbar$ vs. $\hbar$ ) provide a theoretical basis for distinguishing the spin of the radiated particles if the energy loss mechanisms can be disentangled observationally. The study suggests that monitoring binary pulsars (like Hulse-Taylor) could, in principle, test these ratios to confirm the spin-2 nature of gravitons or detect ultralight scalar/vector dark matter.

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