Stochastic Background of Gravitational Waves from Fermions -- Theory and Applications

TL;DR

This work develops a general formalism to compute the stochastic gravitational wave background produced by non-perturbatively excited fermions in the early Universe. It derives the TT source from fermionic anisotropic stress, formulates the unequal-time correlator, and introduces a time-dependent normal-ordering regularization to remove UV divergences, yielding a finite GW spectrum. The authors apply the framework to fermion production during preheating with massless and massive inflatons and to fermion production after reheating (thermal era), deriving the peak frequency k_p and amplitude h^2Ω_{GW}(f_p) as functions of the resonance parameter q and the scalar potential shape; in all cases the predicted GW backgrounds are peaked at very high frequencies, f_p ∼ 10^9–10^{11} Hz, with amplitudes that can be sizable but are generally beyond the reach of planned detectors. The results emphasize that fermions can be efficient GW sources despite Pauli blocking, and they motivate development of ultra-high-frequency GW detectors to probe physics of the very early Universe.

Abstract

Out-of-equilibrium fermions can be created in the early Universe by non-perturbative parametric effects, both at preheating or during the thermal era. An anisotropic stress is developed in the fermion distribution, acting as a source of a stochastic background of gravitational waves (GW). We derive a general formalism to calculate the spectrum of GW produced by an ensemble of fermions, which we apply to a variety of scenarios after inflation. We discuss in detail the regularization of the source, i.e. of the unequal-time-correlator of the fermions' transverse-traceless anisotropic stress. We discuss how the GW spectrum builds up in time and present a simple parametrization of its final amplitude and peak frequency. We find that fermions may generate a GW background with a significant amplitude at very high frequencies, similarly to the case of preheating with scalar fields. A detection of this GW background would shed light about the physics of the very early Universe, but new technology at high frequencies is required, beyond the range accessible to currently planned detectors.

Figures (7)

• Figure 1: Spectrum of GW just at the end of production, see eq. (\ref{['eq:AmpPhi4Prod']}), obtained for parameters $q=10^6$, $h=0.1$ and $\Phi_{I}=\sqrt{{3}/{2 \pi}}M_{p}$, corresponding to an initial energy scale $E_I\approx 6.0\cdot 10^{16} \mathrm{GeV}$. The dashed line has been added by hand, just to help guiding the eye through the scattered points. A realistic spectral shape should be a smooth curve, but due to the rapid oscillations of some of the functions involved, the numerical calculations are not accurate enough. This affects particularly the modes in the UV tail, where the calculated amplitude decrease rather slow. The dashed line plotted lies within the estimated errors from our numerical integration (which we do not show just not to obscure the plot). The 'picture' that one should really take clear from this plot is that there is a well-defined peak in the spectrum, just as expected located at $\kappa \sim q^{1/4}$.
• Figure 2: $q$-dependence of $\mathcal{F}_{*}(\kappa_p)$. The fits correspond to $\mathcal{F}_*(\kappa_p) \propto q^{0.86}$ for $q \geq 1$, and $\mathcal{F}_*(\kappa_p) \propto q^{1.48}$ for $q \ll 1$, see eq. (\ref{['qdepmassless']}).
• Figure 3: The spectrum of GW right after production, see eq. (\ref{['eq:AmpPhi2Prod']}), for parameters $q=10^{6}$, $h=0.1$ and $\Phi_{I}=M_{p}/(2\sqrt{\pi})$, and thus correspondingly for an inflaton mass $m_\varphi=3.4\cdot10^{14} \mathrm{GeV}$ and initial energy density $E_{I}=3\cdot10^{16} \mathrm{GeV}$. Again, the dashed line is simply a guide for the eye over the scattered points. Similar considerations to the ones discussed in the previous section (see the caption of figure \ref{['fig:FermPhi4Spectra']}) apply here as well, in particular what concerns the amplitude of the spectrum at the most UV scales ($\kappa > 5q^{1/4}$).
• Figure 4: $q$-dependence of $\mathcal{F}_{*}(\kappa_p)$. The fits, see eq.(\ref{['qdepmassive']}), correspond to $\mathcal{F}_*(\kappa_p) \propto q^{1.03}$ for $q > 1$, and $\mathcal{F}_*(\kappa_p) \propto q^{2.22}$ for $q < 1$.
• Figure 5: Spectrum of GW right after the end production, for parameters $q=10^{6}$, $m_{\varphi}\approx1.2 \cdot 10^{14}\,\mathrm{GeV}$, initial energy scale $E_I\approx 1.7\cdot10^{16}\,\mathrm{GeV}$ and $\Phi_I=(10^{-2}/h)M_p$. Analogous considerations about the dashed guiding line, similar to those in figures \ref{['fig:FermPhi4Spectra']} and \ref{['fig:FermPhi2Spectra']}, apply here.