[Paper Review] Decay of acoustic turbulence in two dimensions and implications for cosmological gravitational waves

Jani Dahl, Mark Hindmarsh+2 more|arXiv (Cornell University)|Dec 19, 2021

Pulsars and Gravitational Waves ResearchPhysics and Astronomy49 references25 citations

TL;DR

This paper investigates decaying acoustic turbulence in two-dimensional relativistic fluids to model gravitational wave production from cosmological first-order phase transitions. Using numerical simulations under an ultra-relativistic equation of state and non-relativistic velocities, it finds a self-similar energy spectrum with a high-wavenumber power law of k⁻².⁰⁸±⁰.⁰⁸, cut off by shock width, and derives a three-dimensional gravitational wave power spectrum with a primary peak at the initial acoustic scale and a secondary feature at the Hubble-timescale integral scale, enabling improved predictions for LISA-era detectors.

ABSTRACT

Gravitational waves from a phase transition associated with the generation of the masses of elementary particles are within the reach of future space-based detectors such as LISA. A key determinant of the resulting power spectrum, not previously studied, is the lifetime of the acoustic turbulence which follows. We study decaying acoustic turbulence using numerical simulations of a relativistic fluid in two dimensions. Working in the limit of non-relativistic bulk velocities, with an ultra-relativistic equation of state, we find that the energy spectrum evolves towards a self-similar broken power law, with a high-wavenumber behaviour of $k^{-2.08 \pm 0.08}$, cut off at very high $k$ by the inverse width of the shock waves. Our model for the decay of acoustic turbulence can be extended to three dimensions using the universality of the high-$k$ power law and the evolution laws for the kinetic energy and the integral length scale. It is used to build an estimate for the gravitational wave power spectrum resulting from a collection of shock waves, as might be found in the aftermath of a strong first order phase transition in the early universe. The power spectrum has a peak wavenumber set by the initial length scale of the acoustic waves, and a new secondary scale at a lower wavenumber set by the integral scale after a Hubble time. Between these scales a distinctive new power law appears. Our results allow more accurate predictions of the gravitational wave power spectrum for a wide range of early universe phase transition scenarios.

Motivation & Objective

• To model the decay of acoustic turbulence in two-dimensional relativistic fluids following a first-order cosmological phase transition.
• To determine the energy spectrum evolution and shock wave morphology in decaying turbulence.
• To derive scaling laws for kinetic energy and integral length scale decay in the turbulent regime.
• To extrapolate 2D results to three dimensions for estimating gravitational wave power spectra.
• To improve predictions of gravitational wave signals from early universe phase transitions for future space-based detectors like LISA.

Proposed method

• Numerical simulations of a relativistic fluid in two dimensions using a second-order expansion in non-relativistic velocity and viscosity.
• Adoption of an ultra-relativistic equation of state (p = cₛ²ρ) with cₛ = 1, simulating early-universe conditions.
• Initial conditions set as random Gaussian fields with prescribed power spectra to model acoustic waves post-phase transition.
• Use of shock detection algorithms and spectral analysis to characterize energy distribution and decay dynamics.
• Extrapolation of 2D results to 3D using universality of the high-k power law and scaling laws for energy and integral scale.
• Construction of a gravitational wave power spectrum estimate based on shock-induced stress-energy tensor fluctuations.

Experimental results

Research questions

• RQ1 How does the energy spectrum evolve during the decay of acoustic turbulence in 2D relativistic fluids?
• RQ2 What is the functional form of the high-wavenumber tail of the energy spectrum, and what physical scale sets its cutoff?
• RQ3 How do the kinetic energy and integral length scale decay over time in the turbulent regime?
• RQ4 To what extent can 2D simulation results be extrapolated to 3D for gravitational wave power spectrum estimation?
• RQ5 What new spectral features emerge in the gravitational wave power spectrum due to the interplay between initial acoustic scale and Hubble-timescale integral scale?

Key findings

• The energy spectrum evolves toward a self-similar broken power law with a high-wavenumber power law of k⁻².⁰⁸±⁰.⁰⁸.
• The high-k cutoff is set by the inverse width of shock waves, indicating a physical limit to small-scale energy distribution.
• Kinetic energy decays as t⁻¹ and the integral length scale grows as t¹ᐟ², consistent with theoretical expectations for decaying turbulence.
• A transverse kinetic energy component emerges from initially longitudinal-only velocity fields, due to nonlinear fluid dynamics.
• The extrapolated 3D gravitational wave power spectrum features a primary peak at the initial acoustic scale and a secondary feature at a lower wavenumber set by the Hubble-timescale integral scale.
• Between these two scales, a distinctive new power law appears, offering a unique signature for detection in future gravitational wave observatories.