Gravitational waves from stochastic relativistic sources: primordial turbulence and magnetic fields

TL;DR

The paper investigates gravitational waves produced by two stochastic, divergence-free vector sources in the early universe: primordial turbulence and causal magnetic fields. It demonstrates that both sources yield blue spectra on scales larger than their correlation length and develops a framework to compute the resulting GW backgrounds, treating turbulence as an incoherent, short-lived source and magnetic fields as coherent, long-lasting sources. Analytic forms for power spectra, anisotropic stresses, and GW spectra are derived, with explicit dependence on phase-transition parameters and magnetic evolution, and new nucleosynthesis bounds on primordial magnetic fields are established by including sub-horizon modes. The findings show that coherent magnetic fields can be more efficient at generating GW on super-horizon scales, while turbulence generates comparable total GW energy but with distinct spectral features, providing insights for LISA prospects and magnetic-field constraints.

Abstract

The power spectrum of a homogeneous and isotropic stochastic variable, characterized by a finite correlation length, does in general not vanish on scales larger than the correlation scale. If the variable is a divergence free vector field, we demonstrate that its power spectrum is blue on large scales. Accounting for this fact, we compute the gravitational waves induced by an incompressible turbulent fluid and by a causal magnetic field present in the early universe. The gravitational wave power spectra show common features: they are both blue on large scales, and peak at the correlation scale. However, the magnetic field can be treated as a coherent source and it is active for a long time. This results in a very effective conversion of magnetic energy in gravitational wave energy at horizon crossing. Turbulence instead acts as a source for gravitational waves over a time interval much shorter than a Hubble time, and the conversion into gravitational wave energy is much less effective. We also derive a strong constraint on the amplitude of a primordial magnetic field when the correlation length is much smaller than the horizon.

Figures (4)

• Figure 1: The gravitational wave energy density today, normalised to $\left(\mathcal{H}_*L\right)^2 \Omega_{\rm rad}$, as a function of $x=kL$ is shown. We have chosen $v_L^2=\frac{3}{2}\frac{\Omega_T(\eta_*)}{\Omega_{\rm rad}(\eta_*)}=10^{-4}$. The black, oscillating line shows the exact amplitude given in Eq. (\ref{['long']}). The red, straight line shows the approximated spectrum given in Eq. (\ref{['Om0.01']}), normalised to the same quantity $\left(\mathcal{H}_*L\right)^2 \Omega_{\rm rad}$. The factor $2/13$ appearing in Eq. (\ref{['long']}) has been neglected in the plot since it generates an unphysical discontinuity.
• Figure 2: Like Fig. \ref{['fig1']}, but with $v_L=1/2$, hence $\frac{\Omega_T(\eta_*)}{\Omega_{\rm rad}(\eta_*)} = 1/6$.
• Figure 3: The figure shows the gravitational wave energy density today, normalised to $\left(\mathcal{H}_*L\right)^2 \Omega_{\rm rad}$, as a function of $x=kL$, for $v_L=1/\sqrt{3}$, hence $\frac{\Omega_T(\eta_*)}{\Omega_{\rm rad}(\eta_*)}=\frac{2}{9}$ . The red, straight line is the approximation given in Eq. (\ref{['Om3']}).
• Figure 4: The gravitational wave energy density today, normalised to $\Omega_B^2/\Omega_{\rm rad}$, as a function of $x=kL_*$ is shown for $L_*{\cal H}_*=0.01$ in the top panel and for $L_*{\cal H}_*=1$ in the bottom panel. On scales $k<1/\eta_*$ the spectrum behaves as $k^3$, then it flattens into linear growth up to the correlation scale $L_*^{-1}$. On smaller scales it decays like $k^{\alpha+1}$. In this plot $\alpha = -7/2$ and $\gamma=2/7$ are chosen.