Magnetic fields from small-scale primordial perturbations

Weak magnetic fields must have existed in the early Universe, as they were sourced by the cross product of electron density and temperature gradients through the Biermann-battery mechanism. In this paper we calculate the magnetic fields generated at cosmic dawn by a variety of small-scale primordial perturbations, carefully computing the evolution of electron density and temperature fluctuations, and consistently accounting for relative velocities between baryons and dark matter. We first compute the magnetic field resulting from standard, nearly scale-invariant primordial adiabatic perturbations, making significant improvements to previous calculations. This "standard" primordial field has a root mean square (rms) of a few times 10(-15 )nG at 20 less than or similar to z less than or similar to 100, with fluctuations on similar to kpc comoving scales, and could serve as the seed of present-day magnetic fields observed in galaxies and galaxy clusters. In addition, we consider early Universe magnetic fields as a possible probe of nonstandard initial conditions of the Universe on small scales k similar to 1 - 10(3) Mpc(-1). To this end, we compute the maximally allowed magnetic fields within current upper limits on small-scale adiabatic and isocurvature perturbations. Enhanced small-scale adiabatic fluctuations below current cosmic microwave background spectral-distortion constraints could produce magnetic fields as large as similar to 5 x 10(-11) nG at z = 20. Uncorrelated small-scale isocurvature perturbations within current big bang nucleosynthesis bounds could potentially enhance the rms magnetic field to similar to 10(-14) - 10(-10) nG at z = 20, depending on the specific isocurvature mode considered. While these very weak fields remain well below current observational capabilities, our work points out that magnetic fields could potentially provide an interesting window into the poorly constrained small-scale initial conditions of the Universe.