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.

Figures (18)

• Figure 1: Top: Evolution of $\mathcal{A}$ for different initial conditions. Here, each color denotes a different initial ratio $m/H_i$. The dotted, dashed and solid lines correspond to different initializations of the fluctuations with $k_{\rm cut}/m = 1,2,5$. Each curve displays the average of three simulations, with $N^3 = 1250^3$ for $m/H_i = 10$, $20$ and $N^3 = 2048^3$ otherwise. For each simulation, the box size $L$ is chosen such that we are left with one Hubble volume by the end of the simulation and we resolve the DW width by 2 grid points, which corresponds to choosing $\alpha = 1$ and $\beta = 2$ in Eq.\ref{['eq: box size']}. Finally, the gray lines display the result for $\mathcal{A}$ via the VOS model as described in Section \ref{['sec: VOS']}, where in particular we compare to the cases with $k_{\rm cut}/m = 5$. Bottom: Same as above, where the purple curve represents the average of 5 simulations for $N^3 = 2048^3$, $k_{\rm cut}=m$ and $H_i = m$. In this case, and contrarily to the Top plot, the initial DW network is underdense. Here, the box size $L$ was chosen such that $\alpha = \beta = 2$ in Eq.\ref{['eq: box size']}. The gray line displays the area parameter as given by the VOS model.
• Figure 2: Slices of a 3D simulation of a DW network at different times specified by $m/H$ with initial conditions such that $m/H_i = 10$ and $k_{\rm cut}/m = 5$ for $N^3 = 256^3$. The black square indicates the size of one Hubble patch.
• Figure 3: Comparison of the area parameter computed via methods 1 (blue) and 2 (brown) as explained in the text for the initial ratio $m/H_i = 100$ and three different cutoffs $k_{\rm cut}/m = 1$ (dotted), 2 (dashed) and 5 (solid).
• Figure 4: Left: Evolution of the area parameter $\mathcal{A}$ using method 1 as discussed in Section \ref{['sec: scaling numerical results']}. In blue, we show the results for simulations with $N^3 = 2048^3$ grid points, where the averages and error bars were taken from 5 simulations using different base seed values. The same data are also represented by the purple line in Fig. \ref{['fig:approach_scaling']}. Results from the $4K$ simulation are displayed in red. The estimated onset of scaling is around $m\tau = 15$. Right: Evolution of the GW energy fraction $\rho_{\rm gw}/\rho_c$ for the same simulations as in the left plot. The black line corresponds to the expected growth of the GW energy fraction in a RD Universe, i.e.$\sim \tau^4$.
• Figure 5: Top Left: Averaged GW spectra extracted from the same $2K$ simulations as in Fig. \ref{['fig:GW_energy']}, starting from $m\tau = 15$ (bottom red) up to $m\tau = 32$ (top green). The black curve corresponds to the fit on the right plot, while the gray region denotes modes that are omitted from the fit. Top Right: Fit of the spectral shape given in Eq.\ref{['eq: spectral shape']} shown in black from the same $2K$ simulations as in Fig. \ref{['fig:GW_energy']} at the final simulation time (in this case $m\tau_{\rm end} = 32$). The $1\sigma$ error band is visualized by a purple band. Lower Panels: Same as above for one $4K$ simulation at $m\tau = 40$. The purple band represents the $1\sigma$ error band, where the error originates from the fit.