_{1}

^{*}

Recently torsion fields were introduced in CP-violating cosmic axion
a
^{2}
-dynamos [Garcia de Andrade, Mod Phys Lett A, (2011)] in order to obtain Lorentz violating bounds for torsion. Here instead, oscillating axion solutions of the dynamo equation with torsion modes [Garcia de Andrade, Phys Lett B (2012)] are obtained taking into account dissipative torsion fields. Magnetic helicity torsion oscillatory contribution is also obtained. Note that the torsion presence guarantees dynamo efficiency when axion dynamo length is much stronger than the torsion length. Primordial axion oscillations due to torsion yield a magnetic field of 10^{9} G at Nucleosynthesis epoch. This is obtained due to a decay of BBN magnetic field of 10^{15} G induced by torsion. Since torsion is taken as 10^{–20} s^{–1}, the dynamo efficiency is granted over torsion damping. Of course dynamo efficiency is better in the absence of torsion. In the particular case when the torsion is obtained from anomalies it is given by the gradient of axion scalar [Duncan et al., Nuclear Phys B 87, 215] that a simpler dynamo equation is obtained and dynamo mechanism seems to be efficient when the torsion helicity, is negative while magnetic field decays when the torsion is positive. In this case an extremely huge value for the magnetic field of 10^{15} Gauss is obtained. This is one order of magnitude greater than the primordial magnetic fields of the domain wall. Actually if one uses t_{DW} ~ 10^{-}
^{4}
s
one obtains B_{DW} ~ 10
^{22}
G
which is a more stringent limit to the DW magnetic primordial field.

Earlier Mielke and Romero [

Parity violation in gravity has been investigated recently by B Mukhopadhayaya, S Sen and Sur [

violating interactions for spin-

lation in torsion has been used to built dynamo equation. By analogy photon-axion coupling may happen giving rise to magnetic fields that eventually may be amplified giving rise to the

In this section we shall consider the solution the CP-violation dynamos and its efficiency on a torsion back- ground. Let us start by considering the dynamo equation as [

where H is the Hubble parameter, a is the expansion of the universe, S represents the torsion vector and

the metric is given by

Here

From this solution one notes that the torsion oscillating length is complex representing a true oscillation. Here k is the wave coherent scale number. The solution of this equation is

where the oscillation lengths are

Here

The magnetic helicity contribution of torsion oscillation is

Note that the magnetic helicity generation now has a contribution of the torsion oscillation length. Finally let us compute the torsion oscillation new term to compare it with the dynamo length to see if the torsion dissipative term may damp the dynamo length. Then

when the argument is small this expression reduces to

and for Big Bang Nucleosynthesis (BBN)

where we have used the seed field for

which is able to seed galactic dynamos. Dynamo efficiency is given by

where torsion field

Earlier Duncan et al. [

where star in front of the Maxwell tensor F means that we are taking the dual of F given by

By taking the scaled version of this equation in Fourier space one obtains

which solution yields

From this expression one sees that the torsion helicity sign

Here the coupling constant

high as in the next section

Torsion fields introduced in CP-violating cosmic axion

We would like to express my gratitude to D. Sokoloff and A. Brandenburg for helpful discussions on the subject of this paper. I thank Prof. C. Sivaram for initiating me on the problem of dynamos and torsion. Financial support from CNPq. and University of State of Rio de Janeiro (UERJ) are grateful acknowledged.