Revisiting the envelope approximation: gravitational waves from bubble collisions

TL;DR

This work challenges the conventional envelope approximation for gravitational waves from first-order phase transitions by showing its characteristic power-law features are intrinsic and largely independent of bubble nucleation history. It directly compares envelope predictions to large-scale lattice simulations, finding good agreement for scalar-field–driven GW from bubble collisions but notable discrepancies for fluid-driven signals, which are dominated by post-collision acoustic waves. The results delineate the regime where the envelope approach is reliable (scalar-wall collisions) from where it fails (fluid-dominated sources), highlighting the need for new methods to model the fluid contribution in cosmological phase transitions. Overall, the study refines our understanding of GW production mechanisms during phase transitions and informs modeling for future detector forecasts.

Abstract

We study the envelope approximation and its applicability to first-order phase transitions in the early universe. We demonstrate that the power laws seen in previous studies exist independently of the nucleation rate. We also compare the envelope approximation prediction to results from large-scale phase transition simulations. For phase transitions where the contribution to gravitational waves from scalar fields dominates over that from the coupled plasma of light particles, the envelope approximation is in agreement, giving a power spectrum of the same form and order of magnitude. In all other cases the form and amplitude of the gravitational wave power spectrum is markedly different and new techniques are required.

Figures (7)

• Figure 1: Plot comparing (top) the radial field $\phi(\xi)$ and (bottom) fluid velocity $V(\xi)$ profiles for the 'weak' parameters, as a function of $\xi = r/t$ (the parameters can be found in Table \ref{['tab:parameters']}). A weak deflagration with $v_\mathrm{w} \approx 0.44$ is shown. The development of the profiles is illustrated by the red curves (at time intervals of $500/T_\mathrm{c}$ up to $2500/T_\mathrm{c}$), while the profile at $5000/T_\mathrm{c}$ is shown in black. The scalar field bubble wall remains of constant width, while the fluid profile approaches a scaling solution, and is of thickness $\sim \left|v_\mathrm{w} - c_\mathrm{s}\right| R_*/v_\mathrm{w}$ when bubbles of radius $R_*$ collide. The envelope approximation of thin colliding shells might therefore be expected to work for scalar field walls, but plainly cannot for colliding fluid shocks unless $v_\mathrm{w} \approx c_\mathrm{s}$.
• Figure 2: Bubble geometries used in envelope approximation simulations. At left is the widely adopted spherical cutoff, where all gravitational wave power beyond a certain distance from the 'central' bubble is ignored. At right is the 'mirror' approach taken in the present work, where image bubbles are nucleated in neighbouring repeating unit cells; the aim of this is to closely model the periodic boundary conditions of lattice simulations. For a sufficiently large number of bubbles the two approaches are equivalent, corresponding to a system with 'mirror' boundary conditions.
• Figure 3: Comparison of scaled bubble collision power spectra, with $v_\mathrm{w}=1$. We show results from a simulation of 109 bubbles nucleated using the exponentially increasing nucleation rate (squares) and from one where the same number of bubbles are nucleated in the same positions simultaneously (circles). The parameters are such that comparison with Fig. 2 of Ref. Huber:2008hg is also possible, where the bubbles were nucleated at unequal times but with a spherical boundary to the simulation volume. As expected, there is no dependence on the form of the simulation volume. Furthermore, the unequal nucleation time case can be recovered from the equal nucleation time case by the reweighting outlined in the main text (solid blue curve).
• Figure 4: Gravitational waves from colliding scalar field bubble walls. Comparison of power spectra computed by the envelope approximation (points with error bars; blue dashed line fit) and by lattice simulations with source $\tau_{ij}^\phi$ and 'weak' (red curve), 'weak scaled' (purple curve), and 'intermediate' (green curve) phase transition parameters given in the main text. The resulting gravitational wave power is scaled by the scalar field gradient energy density $\kappa^\phi \rho^\text{vac}$, meaning all three lattice simulations are directly comparable to one envelope computation. The envelope computations were made with the same bubble positions, asymptotic wall velocity ($v_\mathrm{w}=0.44$), and scalar gradient energy as develops during the lattice simulation.
• Figure 5: As for Fig. \ref{['fig:unscaled']}, but without the scaling by $\kappa^\phi \rho^\text{vac}$, so comparison with only one lattice simulation is possible, in this case the 'weak scaled' simulation. The box size $L$ and approximate wall width $\ell$ are shown by vertical dashed lines.