We now consider the origin and evolution of cosmological magnetic fields. Our aims are: a) To classify the long list of theories which have been proposed; b) To provide astrophysical arguments constraining these theories, reducing their number, if possible; c) To propose or adopt a coherent overall history of cosmological magnetic fields; d) To determine whether this general scenario provides a reasonable basis for the magnetic hypothesis of rotation curves.
An important argument severely reducing the long list of candidate theories has been given by Lesch and Birk (1998), in which they prove that small scale magnetic fields cannot survive the highly resistive pre-recombination era, characterized by frequent electron-photon interactions. Therefore, cosmic magnetic fields, were either generated after Recombination or had coherence cells larger than the horizon before Recombination. In this latter case, diffusion of magnetic fields cannot proceed over distances larger than the horizon, and these cells could become a subhorizon in the post-Recombination epoch, which is more favourable for the existence of magnetic fields.
Let us then place the limit between the large and small scales at
about 10 Mpc, because this is the minimum length that was superhorizon
before (about) Recombination, or more precisely before "Equality" (the
transition epoch between the radiation and matter
domination). Therefore, magnetic field coherence cells longer than
(today) 
10 Mpc were not set in motion by diffusion in the resistive
era before (about) Equality.
Battaner, Florido and
Jiménez-Vicente (1997)
and Florido and Battaner
(1997)
observed a clear transition in the
evolution of magnetic fields for scales
| (144) |
where mn0 is the present cold matter density (baryonic
or not; dark
or not), 
is the Stephan-Boltzmann constant and
T0 the CMB
temperature. This length is equivalent to just a few Mpc.
This transition is very important for our purposes as large scale fields are not influenced by diffusion or any other micro-physical effect during the radiation dominated era.
Reviews of cosmological magnetic fields have been written by Rees (1987), Coles (1992), Enquist (1978), Olesen (1997), Vallée (1997) and others. Closely related to this topic are the works by Zweibel and Heiles (1997) and by Lesch and Chiba (1997).
In the absence of loss and production amplification mechanisms, frozen-in magnetic field lines will evolve due to the flux of Hubble alone as
| (145) |
where a is the cosmological scale factor, taking its present value as
unity, 
the magnetic field strength when the Universe was
a times smaller than today and 
the present
strength. This equation is in general not true, because the frozen-in
condition is not guaranteed at all epochs, and because production,
loss and amplification processes other than those due to the Hubble
flow could really have taken place. However, we will adopt this
equation, as a re-definition of . The equivalent-to-present
magnetic field strength,
(t), is the strength that would be
observed today corresponding to the real
(t) when
the Universe was a(t)
times smaller, as a result of frozen-in lines in the Hubble flow, in the
absence of an amplifying or destroying mechanism other than the expansion
itself, even
if it does not coincide with the present one at all. As the effect
of expansion is always important, this definition of the
equivalent-to-present magnetic field strength permits a useful
comparison of strengths during different epochs.
On the other hand, this expression can be more general
(Battaner, Florido and
Jiménez-Vicente, 1997)
and holds under the condition of
constituting a small perturbation of the
Robertson-Walker
metrics. A pure U(1) gauge theory with the standard Lagrangian is
conformally invariant (unlike a minimally coupled field), from which
it follows that always decreases in the expansion following
this equation, even in the absence of charge carriers. The Inflation epoch
may be an exception for the reasons given below.
Following Battaner and Lesch (2000), the different theories about the origin of cosmic magnetic fields can be classified into four main groups, characterized by the epoch of formation:
a) during Inflation
b) in cosmological phase transitions
c) during the Radiation Dominated era
d) after Recombination