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Nonlinear Gravity and Multipole Turbulence

A. Ianniccari, A. Kehagias, L. Lo Bianco, A. Riotto

TL;DR

This work derives a kinetic Boltzmann description for the long-time turbulent dynamics of nonlinear gravitational-wave multipoles in flat spacetime, showing that resonant $4$-wave couplings can drive inverse cascades toward lower multipoles. By expanding in multipoles and employing the eikonal limit with Zakharov transformations, the authors obtain stationary power-law spectra that link the exponents to the dispersion and angular-momentum structure, revealing conditions for direct energy cascades and inverse waveaction/ angular-momentum cascades. The analysis yields explicit spectral indices, such as $\nu_E$, $\nu_N$, $\mu_L$, and $\mu_N$, and explains cascade directions via a Fjørtoft-type argument, with implications for the distribution of energy and gravitons across multipoles. The results provide analytic insight into nonlinear GW turbulence, supporting recent numerical findings and suggesting observational consequences for nonlinear QNM signals and early-universe stochastic backgrounds.

Abstract

We derive a kinetic Boltzmann equation characterizing the long-term statistical behavior of the turbulent dynamics of nonlinear interacting gravitational wave multipoles in Minkowski spacetime and show that, injecting a large number of gravitons with large multipoles drives the system toward an inverse multipole cascade at large times.

Nonlinear Gravity and Multipole Turbulence

TL;DR

This work derives a kinetic Boltzmann description for the long-time turbulent dynamics of nonlinear gravitational-wave multipoles in flat spacetime, showing that resonant -wave couplings can drive inverse cascades toward lower multipoles. By expanding in multipoles and employing the eikonal limit with Zakharov transformations, the authors obtain stationary power-law spectra that link the exponents to the dispersion and angular-momentum structure, revealing conditions for direct energy cascades and inverse waveaction/ angular-momentum cascades. The analysis yields explicit spectral indices, such as , , , and , and explains cascade directions via a Fjørtoft-type argument, with implications for the distribution of energy and gravitons across multipoles. The results provide analytic insight into nonlinear GW turbulence, supporting recent numerical findings and suggesting observational consequences for nonlinear QNM signals and early-universe stochastic backgrounds.

Abstract

We derive a kinetic Boltzmann equation characterizing the long-term statistical behavior of the turbulent dynamics of nonlinear interacting gravitational wave multipoles in Minkowski spacetime and show that, injecting a large number of gravitons with large multipoles drives the system toward an inverse multipole cascade at large times.

Paper Structure

This paper contains 2 sections, 120 equations, 1 figure.

Table of Contents

  1. An index for the turbulent cascade: the eikonal limit calculation
  2. The nonlinear time scale

Figures (1)

  • Figure 1: The Fjørtoft argument with $\mu_{TN}=0$, $\mu_{TL}=-1$, $\mu_N=-4/3$ and $\mu_L=-5/3$. For high negative values of the spectral index both fluxes should be positives. Then the total angular momentum flux will be positive constant in $\mu_L=-5/3$, while the flux for the number of graviton will be zero in that point, and so a constant negative in $\mu_N=-4/3$. This argument specifies the directions of both the fluxes: the flux for the number of graviton is negative, therefore the system is moving the number of particles from high $\ell$ to smaller ones. At the contrary, the flux over the angular momentum is positive, so the system will bring the total angular momentum from small values of $\ell$ to higher values.