Cosmic domain walls on a lattice: illusive effects of initial conditions

TL;DR

The paper investigates how cosmic domain-wall networks on a lattice depend on the initial statistical properties of the sourcing scalar field. Using CosmoLattice simulations across biased and unbiased potentials and varying IR/UV content of initial conditions, the authors find that apparent memory of initial conditions arises primarily from lattice boundary effects and IR modes, predicting a universal scaling value $ξ \approx 1.2$ once those artifacts are mitigated. They show that the annihilation time for biased walls follows a $τ_{ann} ∝ ε^{-γ}$ law with $γ ≈ 0.3$ (slightly lower when IR modes are suppressed), implying earlier annihilation than naive estimates. Gravitational-wave predictions inherit a substantial uncertainty (up to a factor of ~5 in $Ω_{gw,peak}$) from initial-condition choices, especially in the low-frequency region, and the post-annihilation GW production in biased walls further shapes the spectrum. Together, these results stress the importance of accounting for IR content and lattice boundaries when interpreting DW dynamics and their observational signatures, including PTA-scale GW backgrounds and future space-based detectors.

Abstract

Evolution of cosmic domain walls (DWs) settles to the scaling solution, which is often assumed to be independent of initial conditions. However, lattice simulations performed in this work reveal a clear dependence of the scaling DW area on the initial configuration of the sourcing scalar field, specifically, its infrared (IR) properties. Namely, the DW area grows as one suppresses IR modes in the initial scalar field spectrum. This growth is saturated, when the area parameter $ξ$ commonly used in the literature reaches the value $ξ_{max} \approx 1.2$. The dependence of $ξ$ on IR modes is argued to be of non-physical origin: it is likely to be due to effects of the lattice boundary. Assuming that physically the memory of initial conditions is erased, one recognizes $ξ\approx 1.2$ obtained in the situation with maximally suppressed IR modes as a genuine universal value of the area parameter in the scaling regime. We demonstrate that ignorance about initial conditions may affect predictions for the energy density of gravitational waves by the factor five. The spectral shape of gravitational waves is also affected by the choice of initial conditions, most notably in the low-frequency part. Likewise, we revisit annihilation of DWs under the influence of a potential bias. It has been previously found in Ref. [19] that the annihilation happens significantly earlier compared to the estimate based on the simple balance between the potential bias and surface energy density. We further support this observation and show that the tendency towards an earlier annihilation gets even stronger upon removing IR modes in simulations.

Figures (11)

• Figure 1: Evolution of the area parameter $\xi$ for different choices of the parameter $\alpha$ characterizing initial conditions in Eq. \ref{['AB']}. An exception is the case $\alpha=0.5$ marked with an asterisk, in which case the r.h.s. of Eq. \ref{['AB']} is multiplied by a factor of $2$, i.e., we have assumed doubled power of the initial scalar spectrum. The IR cutoff is set to $k_{IR}=0$. No UV cutoff is set for $\alpha>0$, while for $\alpha \leq 0$ it is set to $k_{UV}=1$. Simulations have been carried out using $1024^3$ lattice with the box size $L$ given by Eq. \ref{['optimal']}.
• Figure 2: Dependence of the area parameter $\xi$ taken at the conformal time $\tau=15$ on the parameter $\alpha$ characterizing initial conditions in Eq. \ref{['AB']}. The dependence has been inferred from Fig. \ref{['scaling']}.
• Figure 3: Evolution of the area parameter $\xi$ is demonstrated for different values of the cutoff scale $k_{IR}$ parameterizing initial conditions in Eq. \ref{['AB']} with $\alpha=0.5$. No cutoff scale $k_{UV}$ is imposed, unless otherwise specified. The box size has been set either to the optimal side length $L=L_o$ given by Eq. \ref{['optimal']}, to $L_m=L_o/\sqrt{2}$, or to $L_s=L_o/2$. For $L=L_o$ and $L=L_m$, simulations have been performed with $2048^3$ lattice, and for $L=L_s$ they have been performed with $1024^3$ lattice.
• Figure 4: Evolution of the area parameter $\xi$ is demonstrated for various choices of the cutoff scales $k_{IR}$ and $k_{UV}$ parameterizing initial conditions in Eq. \ref{['AB']} with $\alpha=0$. Simulations have been performed with $2048^3$ lattice. The box size has been set to $L=L_o$, given by Eq. \ref{['optimal']}, or $L_m=L_o/\sqrt{2}$.
• Figure 5: Evolution of false vacuum fraction ${\cal F}_{fv}$ under the influence of the potential bias is demonstrated in the range $\epsilon \in [0.0015, 0.05]$. The parameterization \ref{['fvf']} has been used. Simulations have been carried out with $2048^3$ lattice.