Gravitational Waves produced by Compressible MHD Turbulence from Cosmological Phase Transitions

Abstract

We calculate the gravitational wave spectrum produced by magneto-hydrodynamic turbulence in a first order phase transitions. We focus in particular on the role of decorrelation of incompressible (solenoidal) homogeneous isotropic turbulence, which is dominated by the sweeping effect. The sweeping effect describes that turbulent decorrelation is primarily due to the small scale eddies being swept with by large scale eddies in a stochastic manner. This effect reduces the gravitational wave signal produced by incompressible MHD turbulence by around an order of magnitude compared to previous studies. Additionally, we find a more complicated dependence for the spectral shape of the gravitational wave spectrum on the energy density sourced by solenoidal modes (magnetic and kinetic). The high frequency tail follows either a $k^{-5/3}$ or a $k^{-8/3}$ power law for large and small solenoidal turbulence density parameter, respectively. Further, magnetic helicity tends to increase the gravitational wave energy at low frequencies. Moreover, we show how solenoidal modes might impact the gravitational wave spectrum from dilatational modes e.g. sound waves. We find that solenoidal modes greatly affect the shape of the gravitational wave spectrum due to the sweeping effect on the dilatational modes. For a high velocity flow, one expects a $k^{-2}$ high frequency tail, due to sweeping. In contrast, for a low velocity flow and a sound wave dominated flow, we expect a $k^{-3}$ high frequency tail. If neither of these limiting cases is realized, the gravitational wave spectrum may be a broken power law with index between -2 and -3, extending up to the frequency at which the source is damped by viscous dissipation.

Figures (6)

• Figure 1: The left panel shows the Lagrangian eddy turnover time (\ref{['lagtime']}) (red, dashed) and the Eulerian eddy turnover time based on (\ref{['KanSweep']}) (black, solid) and based on the $\langle v_1^2\rangle$ Ansatz (blue, dotted) in arb. units as a function of dimensionless wavenumber $K\equiv kL_I/(2\pi)$. The right panel shows the Gaussian function (\ref{['RSAdec']}) $f_{\rm RSA}$, evaluated with the decorrelation timescales shown in the left panel with corresponding line and color styles at time $\tau=\tau_D$.
• Figure 2: The gravitational wave spectrum for the Higgsportal scenario with $\alpha=0.17$, $\beta/H_*=12.5$, $T_*\approx60$GeV. The lines denote the LISA sensitivity curve (black, solid), the so-far used top hat UTC model (dark-red, dotted), the Lagrangian UTC (blue, dash-dotted) and the Eulerian UTC model (green, dashed). Further we also consider contributions from modes with timescale $\tau_E(k)>\tau_H$$t(k)\gtrsim t_H$ ($\chi=0$). At observable frequencies our calculations based on the sweeping model thus predict an amplitude smaller by roughly a factor 10 compared to the top hat and Lagrangian UTC models. This is mostly due to the shorter correlation timescales in the Eulerian formulation. The two other lines indicate two particular enhancements, the magenta line (dot-dashed) shows the spectrum for the case $\tau_{\rm b}=\beta^{-1}$, whereas the buildup times in the other cases are based on the Eddy turnover time. Lastly, the thick dotted dark-orange line shows the case for maximal magnetic helicity with ($\chi=1$).
• Figure 3: The gravitational wave spectrum for the Higgsportal scenario with $\alpha=0.17$, $\beta/H_*=12.5$, $T_*\approx60$ and maximal magnetic helicity for $\chi=0$ (left panel) and $\chi=2$ (right panel). The lines denote the LISA sensitivity curve (black, solid), the Eulerian UTC without helicity (green, dashed) and different integration times: $\tau_{max}=3\tau_*$ (blue dot-dashed), $\tau_{max}=5\tau_*$ (dark-orange, dot-dashed), $\tau_{max}=10\tau_*$ (brown, double dot),$\tau_{max}=20\tau_*$ (orange-yellow, dotted). The oscillations that appear are generally caused by modes with $\tau_E(k)>\tau_H$ due to the cosine appearing in (\ref{['gwpowspec']}) and also a consequence of the structure function of helical terms $\propto \left(\hat{k}\cdot\hat{p}\right)\left(\hat{k}\cdot\hat{q}\right)$.
• Figure 4: On the left panel, the dependence of the spectrum on the initial value $L_*$ is shown, where $\Omega_{t,*}/\Omega_r=0.2$ ($\alpha\sim0.7$) and $T_*=100$GeV have been chosen. From top to bottom the lines correspond to $L_* H_*=0.4,0.2,0.1,0.05,0.025,0.01,0.005,0.001$ (dark-orange, blue, green, brown, orange, red, dark-red, dark-blue). On the right panel, the dependence on $\Omega_{t,*}$ is investigated for $L_* H_*=0.1$ ($\beta/H_*\sim20$), where from top to bottom the lines correspond to $\Omega_{t,*}/\Omega_r=0.2,0.15,0.1,0.05,0.035,0.025,0.01$ (green, magenta, orange-yellow, brown, blue, dark-red, dark-orange). In both figures and for all scenarios we set $\chi=0$.
• Figure 5: On the left panel, the dependence of the spectrum on the initial value $L_*$ is shown, where $\Omega_{t,*}/\Omega_r=0.2$ ($\alpha\sim0.7$) and $T_*=100$GeV have been chosen. From top to bottom the lines correspond to $L_* H_*=0.4,0.2,0.1,0.05,0.025,0.01,0.005$, where blue lines denote the helical case, while orange lines denote the non-helical case. On the right panel, the dependence on $\Omega_{t,*}$ is investigated for $L_* H_*=0.1$ ($\beta/H_*\sim20$). We show again both the helcial (blue) and nonhelical (orange) scenario. From top to bottom the lines correspond to $\Omega_{t,*}/\Omega_r=0.2,0.15,0.1,0.05,0.035,0.025,0.01$. In both plots we have fixed $\chi=2$ (only modes with $\chi\tau_E(k)<\tau_H$ contribute).