Gravitational Waves sourced by Gauge Fields during Inflation

TL;DR

This work shows that Abelian gauge fields amplified during slow-roll inflation through non-minimal kinetic and axial couplings generate a secondary gravitational-wave background that is scale invariant on super-horizon scales. The authors derive analytic expressions for the gauge-field mode functions, their unequal-time anisotropic-stress spectra, and the resulting GW power spectra, including a full treatment of the evolution through multiple post-inflationary eras. A key result is that a sizeable axial coupling exponentially boosts one GW helicity, yielding a strongly polarized, potentially observable GW signal that can dominate over the standard inflationary background for suitable parameters, while back-reaction constraints limit the allowed region of parameter space. The methodology—combining Whittaker-function solutions, unequal-time correlators, a UV-cutoff consistent with the electromagnetic horizon, and a general multi-era matching formalism—provides a comprehensive framework to predict and constrain gauge-field–sourced GWs in the early Universe with clear observational implications for Planck bounds and next-generation detectors like LiteBIRD.

Abstract

We study the inflationary gravitational wave background induced by Abelian gauge fields generated by non-minimal kinetic and axial couplings to the inflaton. We show that the gravitational wave spectrum is scale invariant and derive its amplitude for generic gauge field coupling parameters, within the slow-roll approximation. We constrain the coupling values and the scale of inflation for which the induced gravitational wave background is observable, while ensuring that back-reaction on the inflationary dynamics remains negligible. We find that a sizeable axial coupling can boost this secondary gravitational wave signal above the standard inflationary background. In the course of our analysis, we also show how to analytically match tensor perturbations across an arbitrary number of eras with different equations of state.

Figures (9)

• Figure 1: The scale independent pre-factors of the symmetric power spectra of magnetic and electric fields as a function of $\gamma_1$. Solid black lines correspond to the exact solution obtained from \ref{['eq:final-whittaker-solution']}; solid red lines are the piecewise approximations \ref{['eq:PB']} and \ref{['eq:PE']}. Although these are not very accurate, they remain much closer to the original solution than the divergent approximation originating from the lowest order term of \ref{['eq:super-horizon-whittaker']}, shown as dotted purple lines. The offset between the black and red lines at $\gamma_1=1$ (for ${\cal P}_\mathcal{B}$) and $\gamma_1=0$ (for ${\cal P}_\mathcal{E})$ is due to having neglected the $\ln(2ik\tau)$ correction in \ref{['eq:Bk-approximation']} and \ref{['eq:Ek-approximation']}. Interestingly, the dependence on $\gamma_1$ is non-monotonic. Meanwhile, the dependence on $\gamma_2$ is simpler and will be discussed in Appendix \ref{['asub:EM-comput']}, see Figure\ref{['fig:anisotropy-spectra']}. Here we have set $\gamma_2=6, |k\tau|=10^{-2}$ and we choose $\delta=0.4$.
• Figure 2: Constraints on $\gamma_1,\gamma_2$ required by the self-consistency relation at the level of the equation of motion, which is the most stringent. Dashed lines show the constrain computed with the approximation \ref{['eq:backreaction-large-g2']} instead of the complete formula. We have considered a slow-roll parameter $\epsilon=0.1$; the inflation energy scale is set at $T_\text{\scriptsize{end}} = 10^{15}$GeV (left plot) and $T_\text{\scriptsize{end}} = 10^{12}$GeV (right plot). $H_\text{\scriptsize{end}}(T_\text{\scriptsize{end}})=\flatfrac{{\cal H}_\text{\scriptsize{end}}}{a_\text{\scriptsize{end}}}$ is given by Eqs. \ref{['eq:a-end']}--\ref{['eq:hub-end']} and is respectively $\simeq 2.2e12\giga\electronvolt$ and $\simeq 2.2e9\giga\electronvolt$.
• Figure 3: The tensor-to-scalar ratio $r(\gamma_1,\gamma_2)$ of gravitational waves induced by gauge fields for different parameters $\gamma_1,\gamma_2$. For the scalar power spectrum we use the Planck values given in Section \ref{['s:GWres']}. The 'spike' at $\gamma_1=1$ is due to a peak in the magnetic field power spectrum that dominates in this region, while the spike at $\gamma_1=0.4$ comes from the electric field, that is somewhat poorly modeled around $\gamma_1=0.4$, see Figure\ref{['fig:electro-fields']}. Near $\gamma_1 =4$ the electric field is diverging, which increases $r$ quite drastically.
• Figure 4: GW energy density per log frequency today, $f=k/(2\pi a_0)$, generated by the inflationary electromagnetic fields. Because of the Gamma and exponential functions in the expression \ref{['eq:final-PT']}, the resulting energy is strongly dependent on the parameter values $\gamma_1,\gamma_2$ (here $T_\text{\scriptsize{end}} =1e15\giga\electronvolt$, hence $H_\text{\scriptsize{end}} \simeq 2.2e12\giga\electronvolt$). All frequencies is the displayed range reenter the horizon deep in the radiation era, resulting in a near scale invariant energy density. We also indicate sensitivity curves from different experiments in gray: 'Square Kilometer Array' (SKA), 'Laser Interferometer Space Antenna' (LISA), 'Einstein Telescope' (ET) and 'Cosmic Explorer' (CE). Data are taken from alberto-roper-pol.
• Figure 5: GW energy density at frequencies around $H_{\rm{eq}}$ for an inflation scale $T_\text{\scriptsize{end}} =1e15\giga\electronvolt$ and several values of $\gamma_1$ and $\gamma_2$ compatible with back-reaction constraints. The frequency dependence is the same as the traditional inflationary tensor spectrum shown in black. The black dot indicates the upper bound from the Planck data at a pivot scale $k_* = 0.002M\mathrm{pc}^{-1}$. This is a continuation of Figure\ref{['fig:omegaGW']} to lower frequencies.