Vector induced Gravitational Waves sourced by Primordial Magnetic Fields

Abstract

In this work, we develop a generic formalism for the study of tensor perturbations induced at second order by first-order vector metric perturbations, dubbing these induced tensor modes $\text{vector-induced gravitational waves}$ (VIGWs). Notably, considering an inflation-inspired power-law type magnetic field power spectrum of the form $P_B(k)\propto k^{n_{\mathrm{B}}}$ (where $n_{\mathrm{B}}$ is the magnetic spectral index), we show that the VIGW signal is enhanced for stiff post-inflationary EoS, with the maximum enhancement happening for $w=1$. We explicitly demonstrate this contribution is dominant over the first-order magnetically-sourced GWs. The VIGW spectrum exhibits a maximum at around the scale crossing the cosmological horizon at the end of reheating, $k_{\mathrm{reh}}$, with its present day peak amplitude scaling as $\hat{I}{\text{c◯}}_{\mathrm{GW}}(k_{\mathrm{reh}},\hat{I}{\cdot }_0)\propto \hat{I}”N_{\mathrm{reh}}\times (H_{\inf }/M_{\mathrm{Pl}}{)}^8$, where $H_{\inf }$ is the Hubble parameter at the end of inflation and $\hat{I}”N_{\mathrm{reh}}$ the duration of the post-inflationary era in $e$-folds. For $w=1$ (kination) and $n_{\mathrm{B}}>-3/2$, one further obtains a nearly $n_{\mathrm{B}}$-independent frequency scaling of the GW spectrum of the form $\hat{I}{\text{c◯}}_{\mathrm{GW}}(f,\hat{I}{\cdot }_0)\propto {\left(\frac{f}{f_{\mathrm{reh}}}\right)}^{-2.8}$ for $f>f_{\mathrm{reh}}\equiv k_{\mathrm{reh}}/(2\ddot{I}€)$. Finally, we highlight that the VIGW signal can be well within the detection bands of several next-generation interferometric GW missions at small scales. Indicatively, for $H_{\inf }\sim O(10^7)\mathrm{GeV}$ and $O(10^{14})\mathrm{GeV}$, and $\hat{I}”N_{\mathrm{reh}}\sim 15$ and $10$, the VIGW signal is expected to be detectable by LISA and ET respectively.

20 pages plus appendices (26 pages in total), 8 figures, version accepted in JCAP with minor text revision