Leptogenesis in the Symmetric Phase of the Early Universe: Baryon
  Asymmetry and Hypermagnetic Helicity Evolution
  We investigate the evolution of the baryon asymmetry of the Universe (BAU) in
its symmetric phase before the electroweak phase transition (EWPT) induced by
leptogenesis in the hypermagnetic field of an arbitrary structure and with a
maximum hypermagnetic helicity density. The novelty of this work is that the
BAU has been calculated for a continuous hypermagnetic helicity spectrum. The
observed BAU $B_{obs} = 10^{-10}$ that can be in large-scale hypermagnetic
fields satisfying the wave number inequality $k \leq k_{max}$ grows with
increasing $k_{max}$. We will also show that the initial right-handed electron
asymmetry $\xi_{eR}(\eta_0)$ used in our leptogenesis model as a free parameter
cannot take too large values, $\xi_{eR}(\eta_0) = 10^{-4}$, because this leads
to a negative BAU by the EWPT time. In contrast, a sufficiently small initial
right-handed electron asymmetry, $\xi_{eR}(\eta_0)$, provides its further
growth and the corresponding BAU growth from zero to some positive
value,including the observed $B_{obs} = 10^{-10}$.
