Gravitational wave production: A strong constraint on primordial magnetic fields

TL;DR

This work examines gravitational wave production from stochastic primordial magnetic fields and uses the nucleosynthesis bound on extra radiation to constrain the magnetic-field amplitude as a function of the spectral index $n$ and creation time $\eta_{in}$. By modeling the fields as a Gaussian random field with a power-law spectrum and computing the resulting anisotropic stresses $\Pi_{ij}$, the authors solve the gravity-wave equation and derive the induced energy-density spectrum $d\Omega_G/d\log k$, highlighting that gravity waves typically impose much stronger limits than CMB anisotropies, especially for $n>-3/2$. The main results show that causally generated fields ($n\ge 2$) must have $B_\lambda \lesssim 10^{-27}$ G, while inflationary scenarios with $n\sim 0$ require $B_\lambda \lesssim 10^{-39}$ G; only very red spectra near $n \approx -3$ are weakly constrained. Consequently, most proposed primordial seed mechanisms for large-scale magnetic fields are strongly disfavored by the gravity-wave bounds, favoring late-time or recombination-era generation, with potential detectability of the gravity-wave background by LISA.

Abstract

We compute the gravity waves induced by anisotropic stresses of stochastic primordial magnetic fields. The nucleosynthesis bound on gravity waves is then used to derive a limit on the magnetic field amplitude as function of the spectral index. The obtained limits are extraordinarily strong: If the primordial magnetic field is produced by a causal process, leading to a spectral index $n\ge 2$ on super horizon scales, galactic magnetic fields produced at the electroweak phase transition or earlier have to be weaker than $B_\la \le 10^{-27}$Gauss! If they are induced during an inflationary phase (reheating temperature $T\sim 10^{15}$GeV) with a spectral index $n\sim 0$, the magnetic field has to be weaker than $B_\la \le 10^{-39}$Gauss! Only very red magnetic field spectra, $n\sim -3$ are not strongly constrained. We also find that a considerable amount of the magnetic field energy is converted into gravity waves. The gravity wave limit derived in this work rules out most of the proposed processes for primordial seeds for the large scale coherent magnetic fields observed in galaxies and clusters.

Figures (2)

• Figure 1: We show $\Omega_Gh_0^2$ and $\Omega_B(\eta_{in})h_0^2$ as functions of the spectral index $n$ for two different times of primordial magnetic field creation: the electroweak transition ( $\Omega_Gh_0^2$ dash-dotted, blue and $\Omega_B(\eta_{in})h_0^2$ short-dashed, red), and inflation ( $\Omega_Gh^2$ dotted, blue and $\Omega_B(\eta_{in})h_0^2$ long-dashed, red) for a fiducial field strength $B_\lambda=10^{-20}$Gauss at $\lambda=0.1$Mpc. The nucleosynthesis limit, $\Omega_{\lim} h_0^2$ is also indicated. (The $\log$- terms have been neglected.) Clearly, the regimes with $\Omega_B>1$ or $\Omega_G>1$ are not physical and are just shown for illustration. We have also shown $\Omega_B(\eta_{nuc})h^2$, the magnetic field density which is simply cut off at the nucleosynthesis damping scale (fat solid line).
• Figure 2: We show the nucleosynthesis limit on $B_\lambda$ (solid line) as function of the spectral index, $n$ together with the limit from gravity waves if the primordial field is produced at the electroweak transition (short-dashed) or during inflation (long-dashed) for $\lambda=0.1$h$^{-1}$Mpc$\simeq 10^{13}$sec.