Generation of gravitational waves from freely decaying turbulence

Extension of the decorrelation velocity at scales larger than the integral scale, following the model given in \cref{eq:Vlarge}. It interpolates smoothly between $\vrms^2/3$ in the inertial range as in \cref{eq:Cv2} and $2\vrms^2/15$ on large scales. These limits are shown with the black dashed lines.

\emph{Left panel}: Slice through simulation ($\mathrm{A}'$) showing the velocity initial conditions in real space. \emph{Right panel}: Same slice as the left panel but after a time $\Delta \tau=20.6 \tauxist$ has elapsed.

\emph{Left panel}: Slice through simulation ($\mathrm{A}'$) showing the velocity initial conditions in real space. \emph{Right panel}: Same slice as the left panel but after a time $\Delta \tau=20.6 \tauxist$ has elapsed.

: Simulation (A), unrescaled velocity power spectrum. : Power spectrum rescaled with \cref{eq:pv_scaled_one_scale}.

: Evolution of the velocity power spectrum from simulation (A). \emph{Left panel}: unrescaled power spectrum. The solid black line is the initial condition, \cref{eq:power-spectrum} and coloured lines show the time evolution from $\tau=\tdevel$ up to $\tau = \tend$ with interval $\Delta\tau = 9.96\, \tauxist$. Lighter colours refer to later times. \emph{Right panel}: power spectrum rescaled according to \cref{eq:pv_scaled_one_scale}. The curves are plotted starting from $\tau-\tdevel\simeq 40\,\tauxist$, at fixed intervals $\Delta \tau = 1.99 \tauxist$. : Caption not extracted

: Evolution of $\vrms^2 \xi^{1+\beta}$.

: Power spectrum rescaled with \cref{eq:Pvxi}, assuming $\beta=3$. : \emph{Left panel}: evolution of $\vrms^2 \xi^{1+\beta}$ for different values of $\beta$ in simulation (A). As shown in \cref{eq:vxigeneral}, this quantity is expected to remain constant, thus indicating that $\beta \simeq 3$. \emph{Right panel}: Power spectrum rescaled with \cref{eq:Pvxi}, using $\beta=3$. The power spectra are plotted from $\tau-\tdevel\simeq 40\,\tauxist$ onward, at fixed interval $\Delta \tau = 1.99 \tauxist$.

Evolution of the velocity and integral scale in simulation (A). The black-dashed line showcases \cref{eq:VelEvMod,eq:XiEvMod} for $\ndecay=5$, $p=4/3$ and $q=1/3$, where the values of $p$ and $q$ correspond to $\beta=3$; see the relationships in \cref{eq:pchialpha,eq:qchialpha}.

: Instantaneous exponents $(p,q)$.

: Time evolution $p(\tau)$, $q(\tau)$. : \emph{Left panel}: Trajectory of the instantaneous exponents $(p,q)$ in simulation (A). Time is represented by the colour scheme: early times are shown with lighter shades and late times with darker shades, starting at $\tau=\tdevel$ and with interval $\Delta \tau \simeq 2 \tauxist$ between markers. The dark solid line represents the scale-invariance line $p=2(1-q)$, \cref{eq:pVsq}. The coloured dashed, dotted and dash-dotted lines show the self-similarity relations $p=(1+\beta)q$ for various choices of $\beta$. \emph{Right panel}: Evolution of the instantaneous kinetic energy and integral scale exponents $(p, q)$ as a function of time in simulation (A). The dash-dotted line shows the values expected for $p$ and long dashed line for $q$, with $\beta = 3$.

: Model of \Refa{kaneda_lagrangian_1993}.

: Our model. : Comparison of normalized unequal time correlator models with data from simulation (A): on the left, the model of \Refa{kaneda_lagrangian_1993} as used in \Refa{Niksa:2018ofa}, on the right our model. The $y$-axis displays the combination of data and model functions which should produce a Gaussian curve in the argument of the $x$-axis; this is discussed in further detail in connection with \cref{eq:Rfigure2b}. The solid dark line is the prediction of the model in each case.

: Simulation (A), $\Delta\tau/\tauxist = 3.98$.

: Simulation (D), $\Delta\tau/\tauxist = 4.87$. : Simulation (E), $\Delta\tau/\tauxist = 0.594$.

: Gravitational wave power spectrum from our simulations with $\vrmsst \approx 0.1$. In the y-axis, we divide the by $(\mathcal{H}_*\xi_*)^2$ since the simulations are in flat space-time, see~\cref{eq:OmGWsimul} and the following discussion. The coloured lines show the gravitational wave power spectrum at intervals $\Delta \tau$ starting from $\tau - \tdevel = \Delta \tau$ and finishing at $\tau = \tend$ . Darker shades correspond to later times. The black dashed line shows an average over the gravitational wave power spectrum in the last half of the elapsed simulation time. The red dashed line shows a $k^1$ power-law, while the blue dashed line shows a $k^{-8/3}$. The sampling of relatively few $k$-space modes into discrete bins leads to some noise in the very lowest wavenumbers. It is also possible that, for these lowest wavenumbers, finite volume effects lead to a flattening of the power spectrum as this is not seen in our results using numerical integration. We have cut off the spectrum at high wavenumbers as it progressively gets polluted due to numerical precision errors in projecting $\dot{u}_{ij}(\vb{k})$ to $\dot{h}_{ij}(\vb{k})$. : Caption not extracted

Averaged GW power spectra for simulations (A)-(G) from \cref{tab:list}. The coloured lines shown here correspond to averaging the GW power spectra over the last half of the simulations. Simulations (B) and (C) have $\vrmsst\simeq0.03$, (A), (D) and (E) have $\vrmsst\simeq0.1$, and (F) and (G) have $\vrmsst\simeq0.3$. A cut off has been applied to the spectrum at high wavenumbers due to numerical precision noise.

Examples of GW power spectra computed via numerical integration of \cref{eq:OmPvAVG} under the assumption of instantaneous turbulence generation (see \eqref{eq:disc_evol}). The spectra are computed using different values of the initial rms velocity, $\vrmsst$, the initial integral scale relative to the Hubble scale, $\mathcal{H}_*\xi_*$. Each panel corresponds to a different value of $\vrmsst$, as specified by the title, and each colour indicates a different value of $\mathcal{H}_*\xi_*$, as specified by the legend in the bottom right panel. The solid lines are computed setting $\beta = 3$, whereas the dashed lines setting $\beta = 4$. We recall that the evolution of $\vrms(\tau)$ and $\xi(\tau)$ are determined by $\beta$ through \cref{eq:pchialpha,eq:qchialpha}.

Anisotropic stress spectral density $P_{\tilde{\Pi}}(k,\tdevel)$ at the initial time in the constant source approximation, from the exact integration of \cref{eq:Piintegral} (black solid line) and the analytical approximation \cref{eq:Piapprox} (red dashed line). We also show $P_{\tilde{\Pi}}(k,\tau)$ extracted from simulation (A) (colored solid lines) between $\tau=\tdevel$ and $\tau=\tend$ with interval $\Delta \tau = 9.96 \tauxist$. Lighter colours indicate later times.

Reproduction of \cref{fig:varying-beta}, where here we also show the constant source approximation given in \cref{eq:constant_approx} for different values of $\vrmsst$ and $\mathcal{H}_*\xi_*$. The solid lines match those of \cref{fig:varying-beta} (with $\beta=3$), whereas the dashed lines give the constant source approximation for an equivalent value of $\vrmsst$ and $\mathcal{H}_*\xi_*$. We fix $\ncut=7$ in the constant source approximation \cref{eq:constant_approx} for all values of $\vrmsst$ and $\mathcal{H}_*\xi_*$.

GW power spectrum for instantaneous turbulence generation. The gray lines show the analytical approximation of \cref{eq:constant_approx} based on a constant source lasting for $\ncut = 7$ eddy turnover times. The black lines show the result of the 4d numerical integration of \cref{sec:numintegration}. From top to bottom, these lines correspond to $\vrmsst=0.3$, $\vrmsst=0.1$ and $\vrmsst=0.03$ respectively. In all cases we fix $\mathcal{H}_* \xi_* = 0.001$. We also show the averaged GW power spectra for simulations (A)-(G) from \cref{tab:list}, which are plotted using colored lines as specified in the legend. The GW power spectra from simulations has been cut off at high wavenumbers due to numerical precision noise.

Gravitational wave power spectrum in the instantaneous generation scenario (solid lines), with a $\mathcal{C}^0$ growth phase (dashed lines) and with a $\mathcal{C}^1$ growth phase (dotted lines). From bottom to top, $\vrmsst = 0.1, 0.3$ and $0.6$. The left panel shows $\mathcal{H}_* \xi_* = 10^{-3}$, the middle panel $\mathcal{H}_* \xi_* = 10^{-2}$ and the right panel $\mathcal{H}_* \xi_* = 10^{-1}$.

Gravitational wave power spectrum in the scenario with $\mathcal{C}^1$ growth phase. Each panel displays a different value for the initial integral scale $\mathcal{H}_* \xi_*$ (as specified in the panels titles), and each line corresponds to $\vrmsst = 0.1, 0.2, 0.3, 0.4, 0.5,$ and $0.6$ from bottom to top.

As in \cref{fig:varying-beta}, we show the SGWB spectra for several values of $\vrmsst$ and $\mathcal{H}_*\xi_*$ in the instantaneous generation scenario with $\beta=3$ (solid lines), compared with the SGWB obtained within the stationary assumption given in \cref{eq:stationary} (dashed lines). We fix $\ncut=7$ as in the constant source approximation, see \cref{eq:SGWBfinconst}.

Velocity power spectra at $\tau=\tend$ in simulations (A), ($\mathrm{A}'$) and ($\mathrm{A}''$).

: Evolution of $\vrms^2 \xi^{1+\beta}$

: Evolution of $\xi$ and $\vrms^2$ : \emph{Left panel}: Evolution of $\vrms^2 \xi^{1+\beta}$ for different values of $\beta$ in simulations (A), ($\mathrm{A}'$) and ($\mathrm{A}''$). As shown in the main text, this quantity should remain constant in freely decaying turbulence, thus indicating that $\beta \simeq 3$. \emph{Right panel}: Evolution of the velocity and integral scale in simulations (A), ($\mathrm{A}'$) and ($\mathrm{A}''$). The black-dashed line showcases \cref{eq:VelEvMod,eq:XiEvMod} for $\ndecay=5$, $p=4/3$ and $q=1/3$, where the values of $p$ and $q$ correspond to $\beta=3$. In both panels, the coloured solid lines refer to simulation (A), dashed lines to ($\mathrm{A}'$) and dotted lines to ($\mathrm{A}''$).

: Instantaneous exponents $(p,q)$.

: Time evolution $p(\tau)$, $q(\tau)$. : \emph{Left panel}: Trajectory of the instantaneous exponents $(p,q)$ in simulation (A), ($\mathrm{A}'$) and ($\mathrm{A}''$). Simulation (A) is shown with circular gray markers, ($\mathrm{A}'$) with pink triangles, and ($\mathrm{A}''$) with orange squares. Time is represented by the colour scheme: early times are shown with lighter shades and late times with darker shades, starting at $\tau=\tdevel$, with interval $\Delta \tau \sim 2 \tauxist$. The dark solid line represents the self-similarity line $p=2(1-q)$. The coloured lines show the relation $p=(1+\beta)q$ for various choices of $\beta$. \emph{Right panel}: Evolution of the instantaneous kinetic energy and integral scale exponents $(p, q)$ as a function of time in simulation (A), ($\mathrm{A}'$) and ($\mathrm{A}''$). Solid lines refer to simulation (A), dashed lines to ($\mathrm{A}'$) and dotted lines to ($\mathrm{A}''$). The horizontal lines show the values expected for $p$ and $q$ if $\beta = 3$.

: Simulation (B).

: Simulation (C). : Real part of the unequal time correlator measured in simulation (B) on the left panel and simulation (C) on the right panel. The $y$-axis displays $\tgauss(k, \tau, \tuetc)$. The solid dark line is the prediction of our model combining \cref{eq:expvsweep,eq:Vlarge,eq:vsweepcomplete}.

: Simulation (D).

: Simulation (E). : Real part of the unequal time correlator measured in simulation (D) on the left panel and simulation (E) on the right panel. The $y$-axis displays $\tgauss(k, \tau, \tuetc)$. The solid dark line is the prediction of our model combining \cref{eq:expvsweep,eq:Vlarge,eq:vsweepcomplete}).

: Simulation (F).

: Simulation (G). : Real part of the unequal time correlator measured in simulation (F) on the left panel and simulation (G) on the right panel. The $y$-axis displays $\tgauss(k, \tau, \tuetc)$. The solid dark line is the prediction of our model combining \cref{eq:expvsweep,eq:Vlarge,eq:vsweepcomplete}).

: Simulation (B), $\Delta\tau/\tauxist = 4.97$.

: Simulation (C), $\Delta\tau/\tauxist = 0.647$. : GW power spectrum from simulations with $\vrmsst \approx 0.03$. The left and right panels show simulation (B) and (C) from \cref{tab:list} respectively. The coloured lines show the GW power spectrum with interval $\Delta \tau$ as listed in the caption, with darker shades corresponding to later times. The black dashed line shows an average over the GW power spectrum in the last $50\%$ of the simulation. We have cut off the spectrum at high wavenumbers due to numerical precision noise.

: Simulation (F), $\Delta\tau/\tauxist = 5.16$.

: Simulation (G), $\Delta\tau/\tauxist = 0.583$. : GW power spectrum from simulations with $\vrmsst \approx 0.3$. The left and right panels show simulation (F) and (G) from \cref{tab:list} respectively. The coloured lines show the GW power spectrum with interval $\Delta \tau$ as listed in the caption, with darker shades corresponding to later times. The black dashed line shows an average over the GW power spectrum in the last $50\%$ of the simulation. We have cut off the spectrum at high wavenumbers due to numerical precision noise.

Evolution of the kinetic energy $\vrms^2$, decomposed into the vortical, $\vrms_\perp^2$, and longitudinal, $\vrms_\parallel^2$, components (see \cref{eq:kinetic-comp}). The central panel shows simulation (A).

: Simulation (B).

: Simulation (C). : Evolution of the kinetic energy $\vrms^2$, decomposed into the vortical, $\vrms_\perp^2$, and longitudinal, $\vrms_\parallel^2$, components (see \cref{eq:kinetic-comp}). The left and right panels show simulation (B) and (C) respectively.

: Simulation (D).

: Simulation (E). : Evolution of the kinetic energy $\vrms^2$, decomposed into the vortical, $\vrms_\perp^2$, and longitudinal, $\vrms_\parallel^2$, components (see \cref{eq:kinetic-comp}). The left and right panels show simulation (D) and (E) respectively.

: Simulation (F).

: Simulation (G). : Evolution of the kinetic energy $\vrms^2$, decomposed into the vortical, $\vrms_\perp^2$, and longitudinal, $\vrms_\parallel^2$, components (see \cref{eq:kinetic-comp}). The left and right panels show simulation (F) and (G) respectively.