Generation of scale invariant magnetic fields in bouncing universes
  We consider the generation of primordial magnetic fields in a class of
bouncing models when the electromagnetic action is coupled non-minimally to a
scalar field that, say, drives the background evolution. For scale factors that
have the power law form at very early times and non-minimal couplings which are
simple powers of the scale factor, one can easily show that scale invariant
spectra for the magnetic field can arise before the bounce for certain values
of the indices involved. It will be interesting to examine if these power
spectra retain their shape after the bounce. However, analytical solutions for
the Fourier modes of the electromagnetic vector potential across the bounce are
difficult to obtain. In this work, with the help of a new time variable that we
introduce, which we refer to as the ${\rm e}$-${\cal N}$-fold, we investigate
these scenarios numerically. Imposing the initial conditions on the modes in
the contracting phase, we numerically evolve the modes across the bounce and
evaluate the spectra of the electric and magnetic fields at a suitable time
after the bounce. As one could have intuitively expected, though the complete
spectra depend on the details of the bounce, we find that, under the original
conditions, scale invariant spectra of the magnetic fields do arise for
wavenumbers much smaller than the scale associated with the bounce. We also
show that magnetic fields which correspond to observed strengths today can be
generated for specific values of the parameters. But, we find that, at the
bounce, the backreaction due to the electromagnetic modes that have been
generated can be significantly large calling into question the viability of the
model. We briefly discuss the implications of our results.
