Cosmic Battery

Let us consider a beam of electrons and protons moving with velocity $v$ with respect to some radiation field with energy density $u$. As a result of Compton scattering, the electrons will feel a drag. Compton drag decelerates the electrons on a timescale

$\tau \approx \frac{m_e c}{r_e^2 u}$

where $m_e$ is the mass of the electron, $c$ is the speed of light and $r_e$ is the classical electron radius. The corresponding timescale for proton is greater by the proton to electron mass ratio $\tau_p \approx \tau m_p/m_e$. For simplicity, we will assume the slow down timescale for protons to be infinite. This separation between protons and electrons gives rise to an electric field that is strong enough to compensate for the Compton drag

$\frac{q E}{m_e} \tau \approx v \Rightarrow E \approx \frac{m_e v}{q \tau}$

where $q$ is the elementary charge.

If this plasma is rotating around a gravitating mass with semi major axis $a$, then according to Faraday's law the growth of the magnetic field is given by

$\dot{B} \approx c E/a \Rightarrow \frac{m_e c v}{q \tau a}$

Over a single orbit, the magnetic field increases by an amount

$\Delta B \approx \frac{m_e c}{q \tau} \approx \sqrt{\frac{r_e^3 u^2}{ m_e c^2}}$

So over a period $t$ the magnetic field grows to

$B \approx N \Delta B$

where $N \approx t v/a$ is the number of orbits.

We can use this expression to estimate the magnetic field in galaxies at the epoch of reionisation. The redshift at the time of reionisation is roughly $z = 6$. The temperature of the CMB at present is roughly $T_p \approx 3 \, \rm K$, so at the time of reionisation it was about $T_r \approx T_p \left(1 + z\right) \approx 20 \, \rm K$. The corresponding energy density is [1]$u \approx 10^{-9} \, \rm erg/cc$. Every revolution, the magnetic field increases by [2]$\Delta B \approx 10^{-25} \, \rm G$, the age of the universe at this point is $t \approx 10^9 \, \rm years$ and the orbital period is $10^{8} \, \rm years$, so the number of orbits is $N \approx 10$ and the net magnetic field is $B \approx 10^{-24} \, \rm G$.