The mean magnetic field of the Universe is zero, if we adopt the Cosmological Isotropy Principle, and therefore it would be random at the larger scales. At smaller scales, there can be coherence cells with different sizes, characterized by a mean field. Coherence cells are usually associated with objects or density inhomogeneities . It could therefore be expected that anisotropies in the CMB, which are associated with density inhomogeneities, might correspond to large magnetic coherence cells. Some anisotropies are larger than 2o, i.e. they are larger than the horizon at Recombination. These larger inhomogeneities were generated before or during Inflation. In the same way, Inflation provides the most natural explanation of field inhomogeneities, as it permits a causal connection between two points with a separation that was until fairly recently (until Equality, approximately) smaller than the horizon.
Turner and Widrow (1988)
were pioneers in analyzing this idea. A cloud
with a present diameter 
, had in the past a size a. However, the horizon evolved independently of
a during
the first phase of Inflation, and was then proportional to
a3/2
during the so called reheating phase, to a2 during the
radiation dominated era, and to a3/2
after Equality (when the radiative and the matter energy
densities became equal). Therefore, a cloud could be
subhorizon before or during Inflation, become superhorizon thereafter
and again be subhorizon at present. This could explain how
points in the cloud at distances further than the horizon
before the present CMB anisotropy are causally connected. In the same way,
magnetic coherence cells, causally connected at Inflation, could have
become superhorizon early and reentered the horizon recently.
Coherence could even be due to electromagnetic waves, as the oscillating electric and magnetic fields, when the wavelength became subhorizon, would have appeared as static fields. Only recently, after around Equality, would conductivity have destroyed large scale electric fields and controlled large scale magnetic fields.
The problem inherent in this theory is that a(t) was
exponential
during Inflation, increasing by a factor of 1021. This would
imply a
decrease in B by a factor of 1042 (perhaps much more),
if Ba2
were a constant, i.e. if the U(1) gauge theory were conformally
invariant. This dilution of magnetic fields must be avoided by some
mechanism. Turner and Widrow considered that the conformal invariance
of electromagnetism is broken through gravitational coupling of the
photon. In this case, the electron would have a mass of only about
10-33eV, and therefore be undetectable. Turner and Widrow
predicted
B0 
5 × 10-10G at scales of about 1 Mpc,
which is really
interesting.
Other authors avoided the complete dilution of primordial fields with
other mechanisms.
Ratra (1992)
considered the coupling of the scalar
field responsible for inflation (the inflaton) and the Maxwell field,
obtaining
B0 10-9G at scales of 5 Mpc. Garretson, Field and
Carroll (1992) invoked a pseudo-Goldstone-boson coupled to
electromagnetism (
B0 < 10-21G at 1 Mpc).
Dolgov (1993)
proposed the breaking of conformal invariance through the "phase
anomaly".
Dolgov and Silk (1993)
considered a spontaneous break of
the gauge symmetry of electromagnetism that produced electrical
currents with non-vanishing curl.
Davis and Dimopoulos (1995) considered a magnetogenesis at the GUT phase transition, but their theory is included here because this transition could have taken place during Inflation (10-11G at galactic scales).
Rather interestingly, when considering the Planck era, the Superstring theory leads to an inflationary pre-Big-Bang scenario which supports some of the theories explained before (Veneziano, 1991; Gasperini and Veneziano, 1993a, b; Gasperini, Giovannini and Veneziano 1995a, b; Lemoine and Lemoine, 1995; etc.) rendering derivations from what were assumptions. In this scenario, the electromagnetic field is deduced to be coupled not only to the metric but also to the dilaton background. COBE anisotropies are the result of electromagnetic vacuum fluctuations, involving scales of the order of comoving 100 Mpc, today. For values of some arbitrary parameters, these models provide large enough values of intergalactic fields, even in the absence of galactic dynamos. They are in fact able to explain a possible equipartition of energy between the CMB radiation and magnetic fields. This pre-Big-Bang scenario is really promising as an explanation of primordial magnetic fields and their connection with CMB.