Fluid perturbations from expanding bubbles in first-order phase transitions

Abstract

We study the power spectrum of the velocity field induced during a first-order phase transition occurring in the radiation-dominated era. We focus on the phase of bubble expansion, assuming that it ends with the onset of the sound-wave regime. The main result we present is a refined template for the velocity spectrum at the beginning of the sound-wave phase, which can be used for studying the resulting anisotropic stresses and gravitational wave production. In particular, we find that the breaks in the velocity spectrum are not associated to the bubble size and the sound shell thickness, as previously proposed, but to the position of the discontinuities. This distinction is particularly relevant for supersonic deflagrations, as it implies that the intermediate slope is more pronounced and the two breaks are more separated when the wall velocity approaches the Chapman-Jouget speed, instead of the sound speed. We also show that the asymptotic branches of the velocity power spectrum are determined by the integral over the single-bubble profiles at large scales, and by the discontinuities of the velocity profiles at small scales. Furthermore, we study the dependence of the two breaks and the intermediate slope on the distribution function of the times of bubble nucleation (exponential and simultaneous). All the results presented in this work have been included in the public Python package CosmoGW.

Figures (26)

• Figure 1: Solid lines illustrate the radial velocity $v_\text{\tiny ip} (\xi)$ (left) and enthalpy $w_\text{\tiny ip} (\xi)$ (right) profiles for different values of the bubble wall velocity $\xi_w$ and the benchmark phase transition strength $\alpha = 0.1$. Vertical black, dashed lines are at the sound speed $c_{\rm s} = \tfrac{1}{\sqrt{3}}$ and at the Chapman-Jouget speed $v_{\rm CJ}$ [see Eq. (\ref{['chapman2']})], indicating the boundaries between subsonic deflagrations, hybrids, and detonations. Dotted lines correspond to the positions $\xi_\text{sh}$ where a shock forms for deflagration solutions [see Eqs. (\ref{['cond_shock']}) and (\ref{['cond_shock_w']})].
• Figure 2: Regions of allowed solutions for the phase transition strength $\alpha$, evaluated at the nucleation temperature and the wall velocity $\xi_w$. The upper bound on $\alpha$ corresponds to $\alpha_+ = \tfrac{1}{3}$, being the largest value allowed for deflagrations (see discussion in App. \ref{['1d_profiles']}). This bound is computed numerically and compared to the analytical fit, $\alpha_{\rm max} \simeq {{1\over3}} (1 - \xi_w)^{-13/10}$Espinosa:2010hh. The boundary between subsonic and supersonic deflagrations is given by the dashed vertical line, indicating the speed of sound, $c_{\rm s} = \tfrac{1}{\sqrt{3}}$. The boundary between supersonic deflagrations (hybrids) and detonations is indicated by the Chapman-Jouget speed, $v_{\rm CJ} \, (\alpha)$ [see Eq. (\ref{['chapman2']})].
• Figure 3: The function ${f'}^2 (z)$ characterizes the spectral density of the velocity field. It is shown, normalized by its asymptotic form in the $z \to 0$ limit, ${f'_0}^2 z^2$ [see Eq. (\ref{['asymptotic_fpz']})], in colored solid lines, evaluated at different wall velocities $\xi_w$ for a benchmark phase transition strength $\alpha = 0.1$. The thick solid lines represent the single or double broken power law fits given in Sec. \ref{['sec:f_template']}. The dotted lines are computed using the corresponding toy models: quadratic profiles for subsonic deflagrations and detonations, and linear-constant profiles for hybrids (see Sec. \ref{['sec:toymodel']}). The positions of the scales $z_1$ and $z_2$, where the function ${f'}^2 (z)$ transitions to its asymptotic forms for $z \to 0$ and $z \to \infty$ respectively, are indicated with vertical lines and computed using the fits provided in Eqs. (\ref{['fit_z1']}) and (\ref{['fit_z2']}). The asymptotic limit $z \to 0$ and the envelope ${f'}^2_{\!\!\!\rm env}$ in the $z \to \infty$ limit [see Eq. (\ref{['fpenv_inf']})] are shown in red solid lines. Their intersection corresponds to $z_{\rm cross}$, defined in Eq. (\ref{['slope_z']}). The type of solution is determined by $\xi_w$, where subsonic deflagrations occur for $\xi_w < c_{\rm s}$, hybrids for $c_{\rm s} \leq \xi_w \leq v_{\rm CJ}$, and detonations for $\xi_w > v_{\rm CJ}$, being $v_{\rm CJ}$ the Chapman-Jouget speed, which takes a value $v_{\rm CJ} \simeq 0.78$ for $\alpha = 0.1$ [see Eq. (\ref{['chapman2']})].
• Figure 4: Same as Fig. \ref{['fp2_fp20']} for hybrids with $\alpha=0.1$ and a bubble wall velocity approaching $v_{\rm CJ} (\alpha) \simeq 0.78$. We find that, as we approach $\xi_w \to v_{\rm CJ}^-$, the second scale $z_2$ becomes progressively larger: this is due to the fact that it is characterized by the inverse distance between discontinuities, $\tilde{\Delta} \xi^{-1}$ (see Eq. (\ref{['fit_z2']}) and the right panel in Fig. \ref{['fig:comparison_z1z2']} for a comparison between $\Delta \xi^{-1}$ and $\tilde{\Delta} \xi^{-1}$). A second scale determined by the inverse fluid shell thickness, $\Delta \xi^{-1}$, would not capture well the $z$ dependence of ${f'}^2(z)$.
• Figure 5: Left panel: Dependence on the bubble wall velocity, $\xi_w$, of the parameters $z_1$ (blue) and $z_2$ (red), given in Eqs. (\ref{['fit_z1']}) and (\ref{['fit_z2']}), that characterize the shape of ${f'}^2 (z)$. The double broken power law structure in the envelope of ${f'}^2 (z)$ arises when $\xi_w \gtrsim \tfrac{1}{2} v_{\rm CJ} (\alpha)$, corresponding to the value at which $z_\text{cross}$ (green), given in Eq. (\ref{['slope_z']}), starts to deviate from $z_1$. The Chapman-Jouget speed, $v_{\rm CJ} (\alpha)$, the speed of sound, $c_{\rm s}$, and $\tfrac{1}{2} v_{\rm CJ} (\alpha)$, are indicated by vertical gray lines. We show $z_1$, $z_2$, and $z_\text{cross}$ for $\alpha = 0.01$ (dotted), $0.05$ (dashed), and $0.1$ (solid). Right panel: Comparison of $\tilde{\Delta} \xi^{-1}$ (blue), $\Delta \xi^{-1}$ (red), and $|\xi_w-c_{\rm s}|^{-1}$ (green) for $\alpha = 0.1$. The choices $\Delta \xi^{-1}$Hindmarsh:2013xzaHindmarsh:2016lnkHindmarsh:2019phvJinno:2020eqgJinno:2022mieCaprini:2024gykCaprini:2024hueCaprini:2024ofd and $|\xi_w-c_{\rm s}|^{-1}$Caprini:2015zloCaprini:2019egzRoperPol:2023bqa have been previously considered in the literature to determine the position of the second spectral break. We show how previous choices deviate from $\tilde{\Delta} \xi^{-1}$ for hybrids. The largest separation between the two scales occurs when $\xi_w \lesssim v_{\rm CJ} (\alpha)$, differing from the usual assumption that it occurs when the fluid shell thickness reaches its minimum at $\xi_w \approx c_{\rm s}$Caprini:2019egzHindmarsh:2019phvHindmarsh:2020hopRoperPol:2023bqa.