Most astrophysical bodies including planets, stars and nearby galaxies are magnetized but the origin of such fields is unknown. Here, a scenario is proposed in which the magnetization of cosmic space is caused by relativistic charged particles (CR for cosmic-rays) produced by Supernova explosion of the first generation of massive stars that are also responsible for the re-ionization of the Universe. Streaming of such particles through the hitherto non-magnetized intergalactic space induces return currents and, hence, electric fields in the cosmic plasma. Owing to resistivity inhomogeneities caused by temperature structure in the cosmic plasma, the electric fields possess a rotational component which sustains Faraday's induction. Magnetic fields thus grow at rate of 10^-18-10^-16 Gauss/Gyr, depending on distance from the CR sources, until the temperature of the intergalactic medium is raised by cosmic reionization. After that the unstable interaction of the magnetic field with the CR currents may produce further amplification by orders of magnitude. Matter accreting onto cosmic structures such as galaxies and clusters of galaxies should thereafter be magnetized to a degree that a concert of processes such as adiabatic compression, galactic dynamo and turbulent amplification can amplify to the observed values.

Figure 1.  Baryonic gas density distribution through a slice across the computational box.

Massive stars characterized by a high emission of ionizing UV photons are the main contributors to process of cosmic re-ionization (Ciardi & Ferrara 2005). At the end of their life they explode producing blast waves that release large numbers of CR protons (Krymsky 1977, Axford et al 1977, Bell 1978, Blandford & Ostriker 1978). Lack of preexisting magnetic fields here is not an issue because such fields can be are easily generated by the star itself and transported in the immediate surrounding by a stellar wind. These particles have a much higher energy and diffusive mean free path compared to thermal particles. Thus they efficiently escape from the parent galaxy, most likely forming a pair of beams due to collimation effects by the galaxy disk.

The CR protons carry a small but important electric current j_c. Quasi-neutrality is maintained by a corresponding return current j_t carried by thermal plasma. Moreover, because of magnetic induction, the return current must nearly balance the CR current at each point locally such that curl(B) = (c/4pi)(j_c+j_t). The CR are collisionless with very long mean free paths whereas the thermal particles, because of their low temperature, have mean free paths shorter than scale lengths of interest. Consequently an electric field E = j_t/sigma is required to draw the return current, where sigma is approximately the Spitzer conductivity, sigma=1.2x10^7 (T/K)^(3/2)sec^-1. If the plasma moves at velocity v, the electric field in the `laboratory' frame is E = - v/c x B+(c/4 pi sigma)curl(B)-c j_c/sigma. The curl of the electric field then produces growth of magnetic field according to Faraday's law

The first term on the right hand side transports the frozen-in field with the plasma and can stretch and amplify an already existing magnetic field. The second term represents diffusive spreading of the field in space. Crucially, in contrast, the final "resistive" term of the equation produces magnetic field in a previously unmagnetised plasma. Temperature inhomogeneities, resulting in variations in conductivity, are naturally present throughout cosmic plasma as a result of structure formation. Since there is no reason why they should be aligned with j_c, cosmic magnetic fields may be generated by the resistive term at a rate dB/dt =|c j_c x grad(sigma^(-1)| ~ cj_{c}/\sigma L_{T} where, L_T = T/|grad(T)|, is the characteristic temperature scale. This resistive process depends sensitively on the plasma temperature through sigma. In particular, it operates efficiently while the intergalactic medium is cold, and it is effectively shut down once reionization raises the intergalactic gas temperature to about 10^4 K.
Observational results (Bowens et al. 2007, Oesch et al. 2009) are used to estimate the CR currents from the star-forming galaxies at the epoch of re-ionization using observations and 30% of the energy released by SN explosions is assumed to be converted into CR particles. The plasma temperature and temperature scale length are extracted from a cosmological simulation of structure formation which includes hydrodynamics, the relevant thermodynamic processes for the diffuse baryonic gas, dark matter and gravity using the methods described in Miniati & Colella (2007a, 2007b). The rate of magnetic field growth around a galaxy of luminosity L is then found to be
 

Fig.1 shows the distribution of the baryonic gas density at z~10 on a two dimensional slice across the simulation box. One recognizes a few high density collapsed structures where gas is rapidly cooling. These are the sites where stars and galaxies form and CRs are eventually produced. However, most of the gas (97\%) is still in the diffuse phase, with a density within a factor a few of the mean value. Accordingly, the normalization in the last Equation has been chosen based on the values of T and L_T in Fig.2. Because fluctuations in T and L_T are somewhat anti-correlated, the magnetic field growth rate is found to fluctuate by only a factor of a few about the value in Eq. above.
When Ohmic heating is included the rate growth rate is reduced to 10^-18--10^-16 Gauss/Gyr. However, further amplification of the magnetic field, which continues when the temperature rises, may be possible through the CR-driven non-resonant instability (Bell 2004, 2005). This instability may amplify the field by a large factor on scales less than 1 kpc, particularly close to galaxies.

Figure 2.  Temperature (top) and its characteristic length scale (bottom) as a function gas density in units of its average value. The vertical bars represent the asymmetric root-mean-squared fluctuations about the mean.