## Author(s)

Dankovsky, I., Ramazanov, S., Babichev, E., Gorbunov, D., Vikman, A.## Abstract

We study domain walls (DWs) arising in field theories where $Z_2$-symmetry is spontaneously broken by a scalar expectation value decreasing proportionally to the Universe temperature. The energy density of such melting DWs redshifts sufficiently fast not to overclose the Universe. For the first time, evolution of melting DWs and the resulting gravitational waves (GWs) is investigated numerically using lattice simulations. We show that formation of closed melting DWs during radiation domination is much more efficient compared to the scenario with constant tension DWs. This suggests that it can be the main mechanism responsible for reaching the scaling regime similarly to the case of cosmic strings. However, the scaling behaviour of melting DWs is observed, provided only that the initial scalar field fluctuations are not very large. Otherwise, simulations reveal violation of the scaling law, potentially of the non-physical origin. The spectrum of GWs emitted by melting DWs is also significantly different from that of constant tension DWs. Whether the system has reached scaling or not, the numerical study reveals a GW spectrum described in the infrared by the spectral index $n \approx 1.6$ followed by the causality tail. We attribute the difference from the value $n=2$ predicted in our previous studies to a finite lifetime of the DW network. Notably, the updated index is still in excellent agreement with the recent findings by pulsar timing arrays, which confirms that melting DWs can be responsible for the observed (GW) signal. We also point out that results for evolution of melting DWs in the radiation-dominated Universe are applicable to constant tension DW evolution in the flat spacetime.

## Figures

Snapshots of melting DW evolution obtained with the $1024^3$ lattice in the case of vacuum initial conditions with $k_{cut}=1$. We use dimensionless units of Eq.~\eqref{redefinitions}.

Snapshots of melting DW evolution obtained with the $1024^3$ lattice in the case of vacuum initial conditions with $k_{cut}=1$. We use dimensionless units of Eq.~\eqref{redefinitions}.

Snapshots of melting DW evolution obtained with the $1024^3$ lattice in the case of vacuum initial conditions with $k_{cut}=1$. We use dimensionless units of Eq.~\eqref{redefinitions}.

Scaling parameter $\xi$ obtained with the $1024^3$ lattice in the case of vacuum initial conditions with the cutoff $k_{cut}=1$. One full simulation is analysed. We use dimensionless units of Eq.~\eqref{redefinitions}.

Histograms showing distribution of melting DWs by the area $S$ in the case of vacuum initial conditions with the cutoff $k_{cut}=1$. The height of each column corresponds to the total area of all DWs in the small range $(S, S+\Delta S)$, where $\Delta S$ is the width of a column. Only one simulation with the $1024^3$ lattice has been been performed for this plots. Dimensionless units of Eq.~\eqref{redefinitions} are used.

Histograms showing distribution of melting DWs by the area $S$ in the case of vacuum initial conditions with the cutoff $k_{cut}=1$. The height of each column corresponds to the total area of all DWs in the small range $(S, S+\Delta S)$, where $\Delta S$ is the width of a column. Only one simulation with the $1024^3$ lattice has been been performed for this plots. Dimensionless units of Eq.~\eqref{redefinitions} are used.

Histograms showing distribution of melting DWs by the area $S$ in the case of vacuum initial conditions with the cutoff $k_{cut}=1$. The height of each column corresponds to the total area of all DWs in the small range $(S, S+\Delta S)$, where $\Delta S$ is the width of a column. Only one simulation with the $1024^3$ lattice has been been performed for this plots. Dimensionless units of Eq.~\eqref{redefinitions} are used.

The same as in Fig.~\ref{Histograms}, but assuming thermal initial conditions (with no cutoff).

The same as in Fig.~\ref{Histograms}, but assuming thermal initial conditions (with no cutoff).

The same as in Fig.~\ref{Histograms}, but assuming thermal initial conditions (with no cutoff).

The same as in Fig.~\ref{Histograms}, but assuming thermal initial conditions (with no cutoff).

Left panel. Scaling parameter $\xi$ obtained with $512^3$ lattice in the case of vacuum initial conditions with the cutoff $k_{cut}=1$ (top), $k_{cut}=6$ (middle), and with no cutoff (bottom). Right panel. Same but assuming thermal initial conditions with the cutoff $k_{cut}=0.3$ (top), $k_{cut}=1$ (middle), and with no cutoff (bottom). One simulation has been performed for each plot. We adopt dimensionless units defined in Eq.~\eqref{redefinitions}.

Evolution of the scaling parameter $\xi$ is shown separately for a long wall (black dots) and closed walls (blue dots) in the case of thermal initial conditions (with no cutoff). Simulations are performed on $1024^3$ lattice. Dimensionless units of Eq.~\eqref{redefinitions} have been used.

Spectrum of GWs produced by the network of melting DWs in the case of vacuum initial conditions with the cutoff $k_{cut}=1$ in dimensionless units defined in Eq.~\eqref{redefinitions}. One simulation is performed on $2048^3$ lattice. Brighter colors correspond to the spectrum at later times. The initial and final conformal times of the simulation are set at $\tau_i=1$ and $\tau = 405$ (shown with the yellow line), respectively. Straight lines demonstrate slopes of GW spectrum in its close-to-maximum infrared and ultraviolet parts at $\tau=405$; corresponding spectral indices are also shown. We have set $\alpha=1$, $\lambda_{\chi}=0.03$, $g_* (T_{sc})=100$, and $T_i \approx 1.3 \cdot 10^{17}~\mbox{GeV}$, when performing simulations. For arbitrary $\alpha$, $\lambda_{\chi}$, $g_* (T_{sc})$, and $T_i$, one should multiply values on the plot by $\alpha^6 \cdot (\lambda_{\chi}/0.03) \cdot (T_i/1.3 \cdot 10^{17}~\mbox{GeV})^2 \cdot (100/g_*(T_{sc}))$, see Eq.~\eqref{peaknumerics}.

The same as in Fig.~\ref{spectrum_vacuum}, but in the case of thermal initial conditions (with no cutoff).

Comparison of GW spectra obtained assuming thermal initial conditions with $2048^3$ and $1024^3$ lattices in the case of equal lattice spacing (left) and different lattice spacings (right). Dimensionless units of Eq.~\eqref{redefinitions} have been used.

Comparison of GW spectra obtained assuming thermal initial conditions with $2048^3$ and $1024^3$ lattices in the case of equal lattice spacing (left) and different lattice spacings (right). Dimensionless units of Eq.~\eqref{redefinitions} have been used.

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