Domain walls in the scaling regime: Equal Time Correlator and Gravitational Waves

Abstract: Domain walls are topological defects that may have formed in the early Universe through the spontaneous breakdown of discrete symmetries, and can be a strong source of gravitational waves (GWs). We perform 3D lattice field theory simulations with CosmoLattice, considering grid sizes $N = 1250$, $2048$ and $4096$, to study the dynamics of the domain wall network and its GW signatures. We first analyze how the network approaches the scaling regime with a constant $\mathcal{O}(1)$ number of domain walls per Hubble volume, including setups with a large initial number of domains as expected in realistic scenarios, and find that scaling is always reached in a few Hubble times after the network formation. To better understand the properties of the scaling regime, we then numerically extract the Equal Time Correlator (ETC) of the energy-momentum tensor of the network, thus determining its characteristic shape for the case of domain walls, and verifying explicitly its functional dependence as predicted by scaling arguments. The ETC can be further extended to the Unequal Time Correlator (UTC) controlling the GW emission by making assumptions on the coherence of the source. By comparison with the actual GW spectrum evaluated by CosmoLattice, we are then able to infer the degree of coherence of the domain wall network. Finally, by performing numerical simulations in different background cosmologies, e.g. radiation domination and kination, we find evidence for a universal ETC at subhorizon scales and hence a universal shape of the GW spectrum in the UV, while the expansion history of the Universe may instead be determined by the IR features of the GW spectrum.

• The paper demonstrates that scaling domain wall networks rapidly evolve and emit gravitational waves with a universal UV spectral index.
• It validates the velocity-dependent one-scale model by extracting consistent equal time correlators across diverse cosmological backgrounds.
• Results reveal that gravitational wave signatures from domain walls remain robust against background variations, offering promising detection prospects.

Domain Walls in the Scaling Regime: Equal Time Correlator and Gravitational Waves

Introduction and Theoretical Framework

Domain walls (DWs) are topological defects arising from the spontaneous breakdown of discrete symmetries in early universe cosmology. Networks of DWs are theorized to contribute substantially to the stochastic gravitational wave (GW) background, providing signatures detectable with current and future GW observatories. The paper explores the non-equilibrium evolution, scaling regime, and GW emission from DW networks through high-resolution 3D lattice field theory simulations, focusing on the extraction and interpretation of the Equal Time Correlator (ETC) and Unequal Time Correlator (UTC) of the energy-momentum tensor.

A key theoretical setup involves a real scalar field $\phi$ with double-well potential, $V(\phi) = \frac{\lambda}{4}(\phi^2 - \eta^2)^2$, yielding DWs upon $\mathbb{Z}_2$ symmetry breaking. The analysis proceeds under expanding universe backgrounds parameterized by the scale factor $a(\tau)$ with different equations of state. Initial conditions mimic the Kibble mechanism via randomized Gaussian fluctuations and momentum cutoffs controlling the initial density of domain walls.

Scaling Regime and Dynamic Approaches

The DW network rapidly evolves post-formation towards the scaling regime, characterized by an $\mathcal{O}(1)$ area parameter $\mathcal{A}$—the number of DWs per Hubble volume—and mildly relativistic wall velocities. Simulations with diverse initializations, variations in $m/H$ and $k_{\text{cut}}$, confirm convergence to scaling within a few Hubble times, independently of initial overdensity. The analysis substantiates the velocity-dependent one-scale (VOS) model as an accurate description for the relaxation and scaling of DW networks.

Figure 1: Top: Evolution of $\mathcal{A}$ across varied initial conditions and cutoffs, confirming rapid approach to scaling independent of initial DW density.

The area parameter is robustly estimated via two numerical methods: a link-based surface counting algorithm and a line-of-sight sign-flip counter, both agreeing within percent-level discrepancies. Slices of lattice simulations reveal rapid reduction from dense to sparse wall configurations, visually demonstrating the scaling transition.

Figure 2: Slices of the domain wall network, displaying morphological simplification and reduction in wall density over cosmological time.

Gravitational Wave Emission in the Scaling Regime

DWs emit GWs efficiently during scaling, with GW energy density evolving as $\rho_{\text{gw}}/\rho_c \propto \tau^4$ in radiation domination, consistent with quadrupolar emission estimates. The GW efficiency parameter $\epsilon_{\text{gw}}$ remains constant after the onset of scaling, with numerical values confirmed against earlier results ($\epsilon_{\text{gw}} \approx 0.54$ for $N=2048^3$ grids).

Analysis of the GW power spectra reveals a characteristic broken power law: a causal IR slope ($\propto k^3$), a pronounced peak at the Hubble scale ($k_{\rm peak}\sim \mathcal{H}$), and a subhorizon UV decay ($k^{-1.3}$). The UV slope, peak position, and amplitude were robust to simulation resolution, with detailed fits presented.

Figure 3: GW spectrum shape and peak scaling across simulation times and resolutions; peak at Hubble scale and consistent $k^{-1.3}$ UV slope.

Comparative analysis across literature supports the universality of the spectrum across various initializations and grid sizes. The IR and UV slopes are stable and match previous state-of-the-art lattice studies.

Equal Time and Unequal Time Correlator Extraction

The ETC, $C^T_{\rm dw}(x,x)$, of the stress tensor serves as a diagnostic for scaling, exhibiting power-law decay $C^T_{\rm dw}(x,x)\propto x^{-q}$ with $q\approx 2.8$ for $x\gg 1$, independent of cosmological background. This scaling is confirmed via direct extraction in $N=2048^3$ and $N=4096^3$ simulations and holds for different expansion dynamics (radiation, kination, exotic equations of state).

Figure 4: The ETC $C^T_{\rm dw}(x,x)$ as a function of self-similar variable $x=k\tau$, demonstrating convergence and scaling for varied cosmological times.

The UTC is modeled via ansätze varying the temporal coherence of the source. Interpolating between incoherent and coherent limits, a partial coherence model ($r\approx 0.4-0.5$ in the exponential suppression term) most accurately reconstructs the simulated GW spectrum, indicating the DW network is neither fully coherent nor incoherent.

Figure 5: UTC structure across the diagonal for varying momentum and interpolation parameter $r$, qualitatively matching observed domain wall coherence properties.

This framework unifies the analytic prediction and simulation results—the UV GW spectrum is ultimately set by the ETC exponent $q$ and its universality across backgrounds.

Cosmology Dependence and Universal GW Spectra

Simulations carried out in kination ($\omega=1$) and an exotic background ($\omega=2/3$) reveal remarkably similar ETC exponents and GW spectral slopes to the radiation case. Scaling parameters ($\mathcal{A}$, $\epsilon_{\rm gw}$) are cosmology dependent, but the UV GW spectral index associated with scaling DWs is insensitive to the expansion history.

Figure 6: Overlay of $C^T_{\rm dw}(x,x)$ for radiation, kination, and exotic cosmologies, evidencing universal power-law decay and spectral shape for subhorizon modes.

Variation in the background expansion primarily affects the IR spectrum (for superhorizon modes) and the amplitude/location of the GW peak post-annealing. Spectra redshift and tilt accordingly for epochs with different $p$ exponents.

Figure 7: Sketch of expected present-day GW spectrum, demonstrating shifting peak locations and IR tilts due to non-standard cosmological epochs.

Phenomenological Implications and Detection Prospects

The results have direct consequences for cosmic GW archaeology. The universality of the UV spectrum ($f^{-1.3}$) for DW networks implies robust predictivity for GW backgrounds. The IR behavior may distinguish epochs of non-standard expansion via altered spectral slopes. Simulated spectra—including amplitude shifts and IR/UV tilting—are placed in the context of forecast sensitivities for LISA, BBO, Einstein Telescope, Cosmic Explorer, and LVK O5.

Figure 8: GW spectrum resolution comparison for 2048^3 vs 4096^3 grids, verifying UV stability of spectral slopes and illustrating numerical artifacts outside resolved regime.

Current and next-generation GW detectors could probe GW backgrounds from DWs in a variety of cosmologies, given that amplitudes remain detectable for domain wall abundances down to percent-level fractions under strong bias scenarios.

Conclusion

Comprehensive lattice simulations demonstrate that DW networks robustly enter the scaling regime, emitting GWs with stable, universal UV spectral features independent of background expansion history. Extraction and analysis of the ETC of the stress tensor enables confident extrapolation between computational and theoretical results, while the partially coherent UTC model captures the observed GW spectrum structure. The study's results imply that GW signatures from scaling domain wall networks encode information on both defect microphysics and expansion history, and the prospect for detection at interferometers is promising. Direct extraction of the UTC and extension to bias-driven network collapse phases represent natural future directions.

Table: Summary of Core Numerical Results

Cosmology	Grid Size	$\omega$	$\tilde{\epsilon}_{\rm gw}$	$b$ (UV Index)	$x_p$ (Peak)	$q$ (ETC Exp.)
Radiation	4096	1/3	$0.35 \pm 0.01$	$1.30 \pm 0.01$	$1.01 \pm 0.02$	$2.903 \pm 0.004$
Kination	2048	1	$0.49 \pm 0.02$	$1.26 \pm 0.03$	$2.16 \pm 0.11$	$2.787 \pm 0.004$
Exotic (2/3)	2048	2/3	$0.41 \pm 0.02$	$1.33 \pm 0.03$	$1.64 \pm 0.09$	$2.804 \pm 0.004$

The amplitude, peak scaling, and ETC exponent are consistent across backgrounds for subhorizon modes, underscoring the universality of domain wall GW signals.

• "Domain walls in the scaling regime: Equal Time Correlator and Gravitational Waves" (2511.16649)