The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition

TL;DR

This work tackles the stochastic gravitational wave background generated by turbulence and magnetic fields produced during a first-order phase transition. It advances the modeling by treating the MHD turbulence as a long-lasting, continuously evolving GW source, incorporating Kraichnan-type time decorrelation and a realistic Von Kármán interpolation for the velocity and magnetic spectra. The authors show that a more physically realistic spectrum reduces the peak GW amplitude by over an order of magnitude compared to earlier estimates, while the peak frequency remains similar; for a strong electroweak transition, the predicted signal could be within LISA’s reach. The study highlights the importance of unequal-time correlations and horizon-imposed Batchelor spectra, and it calls for full numerical relativistic MHD simulations to further refine the predictions.

Abstract

We analytically derive the spectrum of gravitational waves due to magneto-hydrodynamical turbulence generated by bubble collisions in a first-order phase transition. In contrast to previous studies, we take into account the fact that turbulence and magnetic fields act as sources of gravitational waves for many Hubble times after the phase transition is completed. This modifies the gravitational wave spectrum at large scales. We also model the initial stirring phase preceding the Kolmogorov cascade, while earlier works assume that the Kolmogorov spectrum sets in instantaneously. The continuity in time of the source is relevant for a correct determination of the peak position of the gravitational wave spectrum. We discuss how the results depend on assumptions about the unequal-time correlation of the source and motivate a realistic choice for it. Our treatment gives a similar peak frequency as previous analyses but the amplitude of the signal is reduced due to the use of a more realistic power spectrum for the magneto-hydrodynamical turbulence. For a strongly first-order electroweak phase transition, the signal is observable with the space interferometer LISA.

Figures (17)

• Figure 1: The function modeling the time dependence of the source used in Section \ref{['sec:constant']}, Eq. (\ref{['e:ftau']}).
• Figure 2: Comparison between a short and long-lasting source. We plot the function $F(x_{\rm in},x_{\rm fin},\Delta x)$ defined in Eq. (\ref{['constantshort']}) (incoherent case) and Eq. (\ref{['constantco']}) (coherent case) as a function of $x_{\rm in}=kt_{\rm in}$. Blue, solid: long lasting coherent and incoherent cases with $t_{\rm fin}/t_{\rm in}=100$ and $\Delta t=0.01\, t_{\rm in}$. Red, dashed: short lasting coherent and incoherent cases with $t_{\rm fin}/t_{\rm in}=1.01$, $\Delta t=0.01\, t_{\rm in}$. The horizontal lines correspond to the incoherent case.
• Figure 3: For a long-lasting source, comparison between the continuous (solid) and discontinuous (dashed) cases. We plot the function $F(x_{\rm in},x_{\rm fin},\Delta x)$ as a function of $x_{\rm in}=kt_{\rm in}$. The incoherent case is not affected by continuity as shown by the red horizontal line (the solid and dashed lines are superimposed). In the coherent case shown in blue, the slope changes in the continuous case for frequencies $k>\Delta t^{-1}$. The values of the parameters are the same as in Fig. \ref{['fig:factorcoherent']}.
• Figure 4: Comparison between the turbulent velocity power spectrum obtained by intersecting the $k^2$ behaviour at small scale with the inertial range $k^{-11/3}$ behaviour (blue), as done in the literature, with the Von Kármán spectrum (black).
• Figure 5: The normalized velocity power spectrum as a function of wavenumber $K_*$ for different times. Left: the phase in which the turbulence is developing, $0\leq y\leq 1$. Right: the phase of free decay, $y\geq 1$. The Kolmogorov microscale is outside the plot range (c.f. end of section \ref{['sec:howlong']}).