Gravitational wave generation from bubble collisions in first-order phase transitions: an analytic approach

TL;DR

Caprini, Durrer, and Servant present an analytic framework for gravitational waves from bubble collisions in first-order phase transitions, modeling the GW source with a stochastic fluid velocity field rather than the envelope approximation. They derive the anisotropic-stress power spectrum from velocity correlators, compute the GW spectrum including detonation and deflagration regimes, and provide closed-form expressions for the peak frequency and spectrum shape. The amplitude scales with the square of the phase-transition duration-to-Hubble ratio and the kinetic-energy fraction, and the peak frequency is tied to the mean bubble size at the end of the transition. The approach yields results consistent with prior numerical estimates while offering insights into spectrum features and enabling predictions for LISA prospects.

Abstract

Gravitational wave production from bubble collisions was calculated in the early nineties using numerical simulations. In this paper, we present an alternative analytic estimate, relying on a different treatment of stochasticity. In our approach, we provide a model for the bubble velocity power spectrum, suitable for both detonations and deflagrations. From this, we derive the anisotropic stress and analytically solve the gravitational wave equation. We provide analytical formulae for the peak frequency and the shape of the spectrum which we compare with numerical estimates. In contrast to the previous analysis, we do not work in the envelope approximation. This paper focuses on a particular source of gravitational waves from phase transitions. In a companion article, we will add together the different sources of gravitational wave signals from phase transitions: bubble collisions, turbulence and magnetic fields and discuss the prospects for probing the electroweak phase transition at LISA.

Figures (14)

• Figure 1: This figure shows the qualitative profile of the velocity of the broken phase fluid in the frame of the bubble center, for detonations (top panel), planar deflagrations (middle panel) and the approximation given in Eq. (\ref{['velocity']}) (bottom panel). The horizontal axis shows $r/t$ where $t$ denotes the time after bubble nucleation ($t=0$) and $r$ is the distance from the bubble center.
• Figure 2: A schematic drawing of the non-zero velocity region, corresponding to the bubble (for detonations) or to the shock front (for deflagrations).
• Figure 3: This figure shows how the intersection volume $V_i$ changes as a function of the separation between ${\mathbf x}$ and ${\mathbf y}$, $r=|{\mathbf x}-{\mathbf y}|$, where ${\mathbf x}$ and ${\mathbf y}$ are located at the centers of the shells. The upper left, upper right, lower left and lower right plots respectively correspond to $0\leq r \leq R-r_{\rm int}$, $R-r_{\rm int} \leq r \leq 2r_{\rm int}$, $2r_{\rm int}<r<R+r_{\rm int}$ and $R+r_{\rm int}\leq r\leq 2R$. Therefore, this figure does not depict bubble collision (in our approach we do not actually collide bubbles). The shaded volume does not represent the volume of intersection between two different shells, but it accounts for all possible positions of the center of the bubble to which two given points ${\mathbf x}$ and ${\mathbf y}$ belong.
• Figure 4: Velocity power spectrum. The top left panel shows the function $A(K)$ determining the diagonal part of the velocity power spectrum and the fit given in Eq. (\ref{['Ak']}) for different values of $s=v_{\rm int}/v_{\rm out}=r_{\rm int}/R$. The solid lines from top to bottom (red, yellow and pink) are the correct functions and the dashed lines (green, blue and cyan) are the fits for $s=0$, $0.6$ and $0.75$ respectively. The top right panel shows the function $B(K)$ and the fit given in Eq. (\ref{['Bk']}), again for the same values of $s$. The approximations overestimate for $s>0.74$ by about 16%. The lower panel shows again $A(K)$ and $B(K)$ and the fits of Eqs. (\ref{['Ak']}, \ref{['Bk']}) for $s=0.6$. The flatter curve is $A(K)$ and the dashed line is its fit given in Eq. (\ref{['Ak']}). We note the white noise behaviour of $A(K)$ for small values of $K$. The more peaked solid line is $B(K)$ and the dashed line the fit given in Eq. (\ref{['Bk']}). At small $K$, $B(K)$ grows like $K^2$ while $A(K)$ is constant. At large $K$ both functions decay like $K^{-4}$.
• Figure 5: These two figures show the exact integral in Eq. (\ref{['Pint']}) in solid (black), and the fit ${\cal I}(K)$ given in Eq. (\ref{['alk']}) in dashed (red). In the right panel we clearly see the white noise behavior for small values of $K$ and the $K^{-4}$ behavior for large values of $K$, as expected (see discussion above and in Appendix \ref{['Appen:largeandsmall']}).