Gravitational waves from vacuum first-order phase transitions. II. From thin to thick walls

In a vacuum first-order phase transition, gravitational waves are generated from collision of bubbles of the true vacuum. The spectrum from such collisions takes the form of a broken power law. We consider a toy model for such a phase transition, where the dynamics of the scalar field depends on a single parameter $\overline{\lambda }$, which controls how thin the bubble wall is at nucleation and how close to degenerate the vacua are relative to the barrier. We extend on our previous work by performing a series of simulations with a range of $\overline{\lambda }$. The peak of the gravitational-wave power spectrum varies by up to a factor of 1.3, which is probably an unobservable effect. We find that the UV power law in the gravitational-wave spectrum becomes steeper as $\overline{\lambda }\to 0$, varying between $k^{-1.4}$ and $k^{-2.2}$ for the $\overline{\lambda }$ considered. This provides some evidence that the form of the underlying effective potential of a vacuum first-order phase transition could be determined from the gravitational-wave spectrum it produces.

Figure 1

The effect on the potential due to the variation of $\overline{\lambda }$.

Figure 2

The critical profile for a series of potentials with different values of $\overline{\lambda }$.

Figure 3

Field profiles of bubbles when the bubble walls have accelerated up to various $\gamma$ factors. Note that $\gamma =1$ corresponds to the critical bubble profile. (a.) $\overline{\lambda }=0.84$ and (b.) $\overline{\lambda }=0.07$.

Figure 4

Evolution of $\gamma$ as defined in Eq. (30) for a series of values of $\overline{\lambda }$.

Figure 5

The collision of two bubbles of the true vacuum plotted for a thin-wall (a) and thick-wall (b) potential. The $x$ axis corresponds to the line joining the two bubble centers, with $D$ being the separation between bubbles. On the $y$ axis, we plot the time $t$ since the bubbles were nucleated. For both these simulations, the bubbles collide when the Lorentz factors of the bubble walls are $\gamma =4.0$.

Figure 6

The evolution of mean energy densities corresponding to the scalar field for simulations with varying $\overline{\lambda }$.

Figure 7

The power spectrum of the scalar field ${\mathcal{P}}_{\varphi }$. In each plot, darker shades indicate later times. The vertical black dotted line shows the location of $k=2\pi /R_{*}$, whereas the vertical dashed colored line shows the location of $k=M_{\mathrm{b}}$. (a.) $\overline{\lambda }=0.84,N_b=512$ and (b.) $\overline{\lambda }=0.07,N_b=512$.

Figure 8

The power spectrum of the transverse traceless shear stress $T_{ij}^{TT}$. In each plot, darker shades indicate later times. The vertical black dotted line shows the location of $k=2\pi /R_{*}$, whereas the vertical dashed colored line shows the location of $k=M_{\mathrm{b}}$. (a.) $\overline{\lambda }=0.84,N_b=512$ and (b.) $\overline{\lambda }=0.07,N_b=512$.

Figure 9

The power spectrum of the transverse traceless shear stress $T_{ij}^{TT}$ at very late times. In each plot, darker shades indicate later times. The vertical black dotted line shows the location of $k=2\pi /R_{*}$, whereas the vertical dashed colored line shows the location of $k=M_{\mathrm{b}}$. The solid black line shows a power law of $k^3$. (a.) $\overline{\lambda }=0.84,N_b=8$ and (b.) $\overline{\lambda }=0.07,N_b=8$.

Figure 10

Evolution of the gravitational-wave power spectrum for the largest simulations performed for each $\overline{\lambda }$. Each simulation uses a simultaneous nucleation scenario, and the Lorentz factor of the wall of a bubble with diameter $R_{*}$ is ${\gamma }_{*}=4.0$. We plot the power spectra at four different times, early collision phase (a), late collision phase (b), early oscillation phase (c), and later in the oscillation phase (d). The black dashed line gives the result from the envelope approximation [44], the black dot-dashed line gives the prediction from the bulk flow model [44], and the solid black line indicates the previous fit provided in Ref. [64]. The envelope approximation and bulk flow model fits are for an exponential nucleation rate. The vertical dotted line gives the location of $k=2\pi /R_{*}$, whereas the colored dashed lines indicate where $k=M_{\mathrm{b}}$. For each simulation, we shade a region corresponding to $\pm$ the difference in power between our high- and medium-resolution runs. At high wave numbers, the signal is overwhelmed by noise arising from single-precision floating point numerical errors. This noise is identified by comparing a smaller single-precision and double-precision run. We therefore apply a cutoff in the UV at $k=\pi /2\mathrm{\Delta }x$.

Figure 11

The power spectrum of the gravitational-wave energy density parameter for two of the simulations listed in Table 3. In these simulations, the metric perturbations are only turned on after the bubbles have finished colliding, at $t/R_{*}=2.5$. In each plot, darker shades indicate later times. The vertical black dotted line shows the location of $k=2\pi /R_{*}$, whereas the vertical dashed colored line shows the location of $k=M_{\mathrm{b}}$. (a.) $\overline{\lambda }=0.84,N_b=8$ and (b.) $\overline{\lambda }=0.07,N_b=8$.

Figure 12

Evolution of the total gravitational-wave energy density parameter ${\mathrm{\Omega }}_{\mathrm{gw}}$ for a series of $\overline{\lambda }$. These are the simulations listed in Table 3, in which the evolution of $h_{ij}^{TT}$ is only turned on at $t/R_{*}=2.5$. The black dashed line represents a linear fit to the data with slope $\frac{d{\mathrm{\Omega }}_{\mathrm{gw}}}{dt}\sim 0.28(H_{*}{\mathrm{\Omega }}_{\mathrm{vac}}/M_{\mathrm{b}}{)}^2/R_{*}$.

Figure 13

Plot of the values of all the fitting parameters in Eq. (53) for a simultaneous nucleation rate. These have been found using the largest simulation for each $\overline{\lambda }$ in Table 2. We plot how these values vary with time during the simulations. In (a), we show the IR power law $a$; in (b), we show the UV power law $b$; in (c), we plot the peak amplitude ${\tilde{\mathrm{\Omega }}}_{\mathrm{GW}}$; and in (d), we plot the peak frequency $\tilde{k}$. The colored bands show the region corresponding to one standard deviation on the fitting parameters. In each plot, we highlight the prediction for each parameter for an exponential nucleation rate in the envelope approximation by a horizontal dashed black line and in the bulk flow model by a dot-dashed black line.

Figure 14

Variation of the gravitational-wave power spectrum with lattice spacing at $t/R_{*}=8.0$ for $\overline{\lambda }=0.18$ with $N_{\mathrm{b}}=4096$. The black dashed line gives the result from the envelope approximation [44], the black dot-dashed line gives the prediction from the bulk flow model [44], and the solid black line indicates the previous fit provided in Ref. [64]. The vertical dotted line gives the location of $k=2\pi /R_{*}$, whereas the red dashed line indicates where $k=M_{\mathrm{b}}$. At high wave numbers the signal is overwhelmed by noise arising from single-precision floating point numerical errors. This noise is identified by comparing a smaller single-precision and double-precision run. We therefore apply a cutoff in the UV at $k=\pi /2\mathrm{\Delta }x$.

Figure 15

Convergence of the fitting parameters in Eq. (53) calculated at the end of each simulation. We plot how the fitting parameters vary with $\mathrm{\Delta }x/\mathrm{\Delta }{x}_{\mathrm{ref}}$, where $\mathrm{\Delta }{x}_{\mathrm{ref}}$ corresponds to the value of $\mathrm{\Delta }x$ used in Table 2. In (a), we show the IR power law $a$; in (b), we show the UV power law $b$; in (c), we plot the peak amplitude ${\tilde{\mathrm{\Omega }}}_{\mathrm{gw}}$; and in (d), we plot the peak frequency $\tilde{k}$. For the peak amplitude, we also plot a linear fit to the continuum value. In each plot, we highlight the prediction for each parameter by the envelope approximation by a horizontal dashed black line and for the bulk flow model by a dot-dashed black line.

Figure 16

Convergence of the fitting parameters in Eq. (a1) calculated at the end of each simulation. We plot how the fitting parameters vary with $\mathrm{\Delta }x/\mathrm{\Delta }{x}_{\mathrm{ref}}$, where $\mathrm{\Delta }{x}_{\mathrm{ref}}$ corresponds to the value of $\mathrm{\Delta }x$ used in Table 2. In (a), we plot the amplitude of the power spectrum at $k=2\pi /R_{*}$, $A$, and in (b), we show the UV power law $b$.

Figure 17

In the top plots, we show the evolution of the bubble radius parameters $r_{\mathrm{mid}}$, $r_{\mathrm{in}}$, and $r_{\mathrm{out}}$ (defined in Sec. 2b) for an isolated bubble. These are given for 1D simulations with various lattice spacings as well as the theoretical behavior. The bottom panels give the fractional deviation from the theoretical value for each lattice spacing. We also include the result of an isolated bubble left to expand in a 3D simulation. (a.) $\overline{\lambda }=0.84$ and (b.) $\overline{\lambda }=0.07$.

Figure 18

Deviation of the bubble wall Lorentz factor $\gamma$ from its theoretical value in 1D simulations of isolated bubbles for a variety of lattice spacings. We also include the result of an isolated bubble left to expand in a 3D simulation. (a.) $\overline{\lambda }=0.84$ and (b.) $\overline{\lambda }=0.07$.

Figure 19

Slices $(0,y,z)$ for a simulation with $\overline{\lambda }=0.07$ and $N_{\mathrm{b}}=64$. In the top row, we plot the scalar field normalized by the broken phase value. The middle row shows the energy density in gravitational waves ${\rho }_{\mathrm{gw}}$. The bottom row shows the modulus of the transverse traceless shear-stress.

Figure 20

Slices $(0,y,z)$ for a simulation with $\overline{\lambda }=0.84$ and $N_{\mathrm{b}}=64$. In the top row, we plot the scalar field normalized by the broken phase value. The middle row shows the energy density in gravitational waves ${\rho }_{\mathrm{gw}}$. The bottom row shows the modulus of the transverse traceless shear stress.