Helical cosmological magnetic fields from extra-dimensions
  We study the inflationary generation of helical cosmological magnetic fields
in a higher-dimensional generalization of the electromagnetic theory. For this
purpose, we also include a parity breaking piece to the electromagnetic action.
The evolution of extra-dimensional scale factor allows the breaking of
conformal invariance of the effective electromagnetic action in $1+3$
dimensions required for such generation. Analytical solutions for the vector
potential can be obtained in terms of Coulomb wave-functions for some special
cases. We also present numerical solutions for the vector potential evolution
in more general cases. In the presence of a higher-dimensional cosmological
constant there exist solutions for the scale factors in which both normal and
extra dimensional space either inflate or deflate simultaneously with the same
rate. In such a scenario, with the number of extra dimensions $D=4$, a scale
invariant spectrum of helical magnetic field is obtained. The net helicity
arises, as one helical mode comes to dominate over the other at the
superhorizon scales. A magnetic field strength of the order of $10^{-9}$ $G$
can be obtained for the inflationary scale $H\simeq 10^{-3}$ $M_{pl}$. Weaker
fields will be generated for lower scales of inflation. Magnetic fields
generated in this model respects the bounds on magnetic fields by Planck and
$\gamma$-ray observations (i.e. $10^{-16}$ $G$ $<$ $B_{obs}<3.4\times 10^{-9}$
$G$).
