Gravitational Wave Signature of Aspherical Bubbles Driven by Thermal Fluctuation

Abstract

Cosmological first-order phase transitions are a well-motivated source of stochastic gravitational waves (GWs), but most predictions are made based on the highly idealized model of perfectly spherical vacuum bubbles, neglecting thermal fluctuations. In this work we use $(3+1)$-dimensional lattice simulations of a scalar model with thermal initial conditions to quantify how thermal fluctuations distort bubble profiles and modify the resulting GW spectrum. We find that thermal fluctuations can strongly break spherical symmetry at early times, allowing even an isolated bubble to emit GWs. In multi-bubble simulations, thermal fluctuations systematically reshape the spectrum, suppressing the infrared part while enhancing and broadening the high-$k$ tail. We further provide an analytical estimate for the ultraviolet regime of the GW spectrum, which is in good agreement with our lattice results and suggests that this regime is dominated by thermal fluctuations. These effects could leave observable imprints in future GW searches.

Figures (8)

• Figure 1: Slices of $\phi$ along the $z$-direction at selected times for the fluctuation-free case (left) and the case with thermal fluctuations at $T = 50~\mathrm{GeV}$ (right). The bubble is initialized on a $256^3$ lattice at $t=0$. For visual clarity, the colorbar is clipped to $[0,1]$.
• Figure 2: Time evolution of the symmetry factor $\epsilon$ computed on a $256^3$ lattice for different temperatures. The purple curve shows the no-fluctuation case, while the light blue, blue, and red curves correspond to fluctuations at $T = 50~\mathrm{GeV}$, $55~\mathrm{GeV}$, and $60~\mathrm{GeV}$. The shaded gray region marks the interval influenced by bubble self-collisions arising from periodic boundary conditions.
• Figure 3: GW spectra computed at $\bar{t} = 30$ for a single bubble with thermal fluctuations at $T = 50~\mathrm{GeV}$. The blue curve shows the result from evolving the full scalar field, the red curve corresponds to the evolution of fluctuations alone, and the orange curve denotes their difference, which can be interpreted as the contribution from the evolution of a non-spherical bubble.
• Figure 4: GW spectra computed on a $512^3$ lattice for multi-bubble evolution, evaluated at $\bar{t} = 4 R_{\mathrm{sep}} M$, for $T = 50~\mathrm{GeV}$, $55~\mathrm{GeV}$, and $60~\mathrm{GeV}$ (blue, green, and orange). Dashed lines denote the corresponding spectra without thermal fluctuations. The vertical dash–dot lines mark the characteristic scales $kR_{\mathrm{sep}} = 2\pi R_{\mathrm{sep}}T$.
• Figure 5: Detectability of the GW spectra extrapolated from simulations performed at $T= 60~\mathrm{GeV}$ and $\bar{t}=4R_{\mathrm{sep}}M$. The solid curves correspond to $HR_{\mathrm{sep}}=0.003$, while the dashed curves denote $HR_{\mathrm{sep}}=0.0001$. The sensitivity curves of LISA, Taiji, TianQin, DECIGO Schmitz:2020syl, and BBO Schmitz:2020syl are shown for comparison. The sensitivity curves for GW experiments are for 1 year observation time, with signal-to-noise ratio = 1.