[Paper Review] Near-Peak Spectrum of Gravitational Waves from Collapsing Domain Walls

Bryce Cyr, Steven J. Cotterill+1 more|ArXiv.org|Apr 2, 2025

Geophysics and Gravity Measurements3 citations

TL;DR

The paper uses 2048^3 lattice simulations of biased Z2 domain walls in a radiation-dominated background to study gravitational wave spectra during scaling and collapse, showing a near-peak spectrum strongly influenced by finite decay time and bias form.

ABSTRACT

Cosmological domain walls appear in many well-motivated extensions to the standard model of particle physics. If produced, they quickly enter into a self-similar scaling regime, where they are capable of efficiently sourcing a stochastic background of gravitational waves. In order to avoid a cosmological catastrophe, they must also decay before their enormous energy densities can have adverse effects on background dynamics. Here, we provide a suite of lattice simulations to comprehensively study the gravitational wave signatures of the domain wall network during this decay phase. The domain walls are initially formed through spontaneous breaking of a $\mathbb{Z}_2$ symmetry, and subsequently decay through the action of a small bias term which causes regions of false vacuum to collapse. We find that gravitational waves are produced in abundance throughout this collapsing phase, leading to a shift in the peak frequency and increase in the overall amplitude of the spectrum by an $\mathcal{O}(100)$ factor when compared against simple analytic arguments. Importantly, we also find that the characteristic frequency of emitted gravitational waves increases as the network decays, which leads to a softening of the high frequency spectral index. This high frequency spectrum therefore carries key information related to the dynamics of the collapsing phase, and can be used to discriminate between different domain wall scenarios using upcoming data.

Motivation & Objective

• Investigate gravitational wave signatures from biased domain wall networks during decay in a radiation-dominated Universe.
• Quantify how bias type and decay timescale affect the near-peak GW spectrum.
• Characterize how the high-frequency spectral index informs the dynamics of the collapsing phase.
• Provide a parameterized framework to connect bias amplitude and decay times to the GW spectrum for comparison with observations.

Proposed method

• Perform large-scale lattice simulations of a real scalar field with Z2 symmetry on a 2048^3 grid in an expanding background.
• Explore two bias forms: a vacuum bias present at all times and a time-dependent (temperature-like) bias that turns on at a critical redshift.
• Compute the gravitational wave spectrum throughout damping, scaling, and decay phases using a quadrupole-based estimate and a volume-normalized spectral function S_k(t).
• Extract the area parameter A from the PRS linking formula to diagnose scaling vs decay regimes.
• Fit the area parameter evolution to A(τ̄)=A_scale exp[-(τ̄/τ̄_dec)^p] to determine decay times and compare with instantaneous-decay predictions.
• Compare decay times with the instantaneous decay condition H(τ̄_b)=V_b/σ to assess lifetime and GW implications.

Experimental results

Research questions

• RQ1 How does a finite domain-wall decay time, as opposed to instantaneous decay, modify the peak frequency and amplitude of the gravitational wave spectrum?
• RQ2 How do different bias implementations (vacuum bias vs. time-dependent bias) affect the lifetime of the domain-wall network and the resulting GW spectrum?
• RQ3 What is the high-frequency spectral behavior during collapse, and what information does it carry about the collapsing dynamics?
• RQ4 Can the near-peak GW spectrum distinguish between domain-wall scenarios using upcoming gravitational wave data?

Key findings

• The near-peak GW spectrum is enhanced by approximately an O(100) factor relative to instantaneous-decay expectations.
• The peak frequency of emitted GWs decreases (redshifts) as the network decays, compared to the instantaneous decay case.
• The high-frequency spectral index softens during decay, with its value tied to the amplitude and time dependence of the bias.
• The lifetime of the biased wall network can be significantly longer than the instantaneous decay time τ̄_b, with fits showing decay times τ̄_dec that exceed τ̄_b by factors up to ~2–3 for studied biases.
• For vacuum bias cases, the study reports A_scale ≈ 0.78 ± 0.03 and p ≈ 4.15–4.33 depending on bias strength, with decay times τ̄_dec ranging from ~12 to ~32 (in bar units) as ε/λ decreases.
• The instantaneous decay approximation underestimates the true lifetime and overestimates how early the network ends radiating gravitational waves.